Place six toy ladybirds into a box so that there are two ladybirds in every column and every row.

Reduce 1,000,000 to 7 on a calculator using only the 7 key and basic arithmetic operations.

Arrange numbers 1 to 8 in circles to make four correct calculations.

Balance an equalizer with four weights.

Change the color of all spots on a grid in fewer throws of a dice, with three versions of the challenge.

Place numbers 1 to 10 in circles such that each number is the difference between the two numbers below it.

Determine the number of ways to hang three weights on one side of a balance to counterbalance two weights on the other side.

Decipher a code based on two prime numbers less than 10, multiplied together and shifting the alphabet forward by the result.

Find the number of quarter turns a man must rotate to match each picture using interactivity.

Investigate why 2 by 2 could be 5 and predict the result of 2 by 10.

Play a version of the classic Countdown Game.

Create a large triangle with small triangles adding up to 10.

Arrange numbers 1 to 15 on circles such that no consecutive numbers lie on a continuous straight line.

Solve the frogs and toads challenge by swapping them in as few slides and jumps as possible.

Determine the chance of winning a lottery played in a far-away land.

Play a train game for an adult and child, where the goal is to make a train first.

Find the number of right angles that can be made using two sticks.

Develop a strategy to stop an opponent from splitting piles of counters into unequal numbers.

Explore the shapes that can be made with overlapping circles of the same and different sizes.

Predict the color of the 18th and 100th marks on a wheel with regular intervals.

Find all combinations of three beads threaded on a circular wire, colored red or blue.

Create an Islamic star and cross design using interactivity, and produce a tessellation of regular octagons with two types of triangles.

Experiment with a tricky tessellation in an interactive activity.

Play a game where a rectangle with 50 squares is filled by rolling a dice to get a percentage between 2 and 100.

Arrange counters on a grid to make rectangles, and determine which numbers of counters can form rectangles and how many different rectangles can be made.

Play a game for two people that is well-known, where the goal is to never lose.

Play the Incey Wincey Spider game, where the goal is to get to the top of the drainpipe.

Arrange four number cards on a grid to make a diagonal, vertical, or horizontal line according to the rules.

Find the coordinates of colored dots marking out a tangram, and explore how changing the origin affects the coordinates.

Exchange coins in a bank to make the same total as given groups, using coins with different spot values.

Play the First Connect Three game, using dice numbers and addition or subtraction to get three numbers in a straight line.

Investigate the possibility of placing counters on a 3x3 grid to have an even number of counters in every row and column.

Find all ways to line up Cuisenaire rods.

Use interactive dominoes to explore different arrangements.

Choose 13 spots on a grid, determine the scoring system, and find the maximum possible score.

Place numbers 1 to 6 in circles such that each number is the difference between the two numbers below it.

Make a train the same length as Laura's using three differently colored rods, and determine if there is only one way to do it.

Solve a 100-square jigsaw written in code, starting with 1 and ending with 100.

Move three circles to make a triangle face in the opposite direction.

Make a dice game fair using interactivity or playing the game.

Use interactivity to find all possible ways balls could land in a problem.

Balance an equalizer by putting more than one weight on a hook.

Work out the times table and shift used by a computer in an activity.

Determine how to make a dark green rod using yellow and white rods.

Find the quadrilaterals that can be made with wooden planks of different lengths for a rabbit run.

Multiple choice questions and answers have been refined for clarity and concision, but none were present in the original text.


# Multiple Choice Questions on Statistics

6. In testing a hypothesis using the X2 test, the theoretical frequencies are based on the
a. null hypothesis.
b. alternative hypothesis.
c. normal distribution.
d. None of the above.
Answer: a. null hypothesis.

7. If we wish to determine whether there is evidence that the proportion of successes is higher in group 1 than in group 2, the appropriate test to use is
a. the Z test.
b. the X2 test.
c. Both of the above.
d. None of the above.
Answer: a. the Z test.

8. The Y-intercept (b0) represents the
a. estimated average Y when X = 0.
b. change in estimated average Y per unit change in X.
c. predicted value of Y.
d. variation around the sample regression line.
Answer: a. estimated average Y when X = 0.

9. If the Durbin-Watson statistic has a value close to 4, which assumption is violated?
a. Normality of the errors.
b. Independence of errors.
c. Homoscedasticity.
d. None of the above.
Answer: b. Independence of errors.

10. The coefficient of determination (r2) tells us
a. that the coefficient of correlation (r) is larger than 1.
b. whether r has any significance.
c. that we should not partition the total variation.
d. the proportion of total variation that is explained.
Answer: d. the proportion of total variation that is explained.

11. The coefficient of multiple determination r2Y.12
a. measures the variation around the predicted regression equation.
b. measures the proportion of variation in Y that is explained by X1 and X2.
c. measures the proportion of variation in Y that is explained by X1 holding X2 constant.
d. will have the same sign as b1
Answer: b. measures the proportion of variation in Y that is explained by X1 and X2.


**Staff Selection Commission – Combined Graduate Level Examination** is an examination conducted to recruit staff to various posts in ministries, departments, and organisations of the Government of India. It is conducted by the Staff Selection Commission for selecting staff for various Group B and Group C posts. The Staff Selection Commission was established in **1975**.

Candidates applying for the various posts need to have a bachelor’s degree from a recognised university at the time of applying. Age of the candidate must be between 18 and 32 years (depending upon post applied).

**Q.1)** *How many numbers are there from 300 to 650 which are completely divisible by both 5 and 7 ?*

a) 8

b) 9

c) 10

d) 12

## Answer

Correct Answer:** (c)** 10

## Solution

The LCM of 5 and 7 is 35.

^{300}⁄_{35} = 8.57 (approx.),

∴ 9 × 35 = 315 which is smallest number divisible by both 5 and 7.

^{650}⁄_{35} = 18.57,

∴ 18 × 35 = 630 which is largest number divisible by both 5 and 7.

Add the digits 9, 10,…….,17 & 18. We got 10 numbers.

**Answer:** There are **10 numbers** between 300 and 650 which are divisible by both 5 and 7.

**Q.2)** *Raman can do a work in 5 days, Jatin can do the same work in 7 days and Sachin can do the same work in 9 days. If they do the same work together and they are paid Rs 2860, then what is the share (in Rs) of Raman ?*

a) 1260

b) 700

c) 900

d) 870

## Answer

Correct Answer:** (a)** 1260

## Solution

*L.C.M. of 5, 7 and 9 = 315*

*Raman works = ^{315}⁄_{5} = 63 units*

*Jatin works = ^{315}⁄_{7} = 45 units and *

*Sachin works = ^{315}⁄_{9} = 35 units.*

*If they worked together, then total work = (63 + 45 + 35) = 143 units*

*According to the problem, we have *

* 143 units = Rs. 2860*

* ∴ 63 units = Rs. ^{(2860 × 63)}⁄_{143} = Rs. 1260 ………(Ans.)*

**Q.3)** *If the diameter of a sphere is 14 cm, then what is the surface area (in cm ^{2}) of the sphere ?*

a) 616

b) 308

c) 462

d) 636

## Answer

Correct Answer:** (a)** 616

## Solution

*Radius of the sphere (r) = ^{14}⁄_{2} = 7 cm.*

*∴ Surface area = 4πr ^{2} *

* = 4 × ^{22}⁄_{7} × 7^{2}*

* = 4 × 22 × 7*

* = 616 sq. cm. ……….(Ans.)*

**Q.4)** *After two successive discounts of 20% and 35%, an article is sold for Rs 50700. What is the marked price (in Rs) of the article ?*

a) 92500

b) 98500

c) 97500

d) 94500

## Answer

Correct Answer:** (c)** 97500

## Solution

*If x be the 1 ^{st} discount and y be the 2^{nd} discount, then we have*

*Total successive discount = *

( x+y− x×y 100 )%

*In this problem, x = 20 and y = 35*

*∴ Total discount = *

( 20+35− 20×35 100 )%

* = 48 %*

*Given, Selling Price (S.P.) of an article = Rs. 50700*

*∴ Market Price (M.P.) = Rs. ^{100}⁄_{(100 − 48)} *

* = Rs. ^{100}⁄_{52} × 50700 *

* = Rs. 97500 …………….(Ans.)*

**Q.5)** *Rs 3200 is divided among A, B and C in the ratio of 3 : 5 : 8 respectively. What is the difference (in Rs) between the share of B and C ?*

a) 400

b) 600

c) 800

d) 900

## Answer

Correct Answer:** (b)** 600

## Solution

*Ratio of shares of A, B and C = 3 : 5 : 8*

*Sum of ratio = 3 + 5 + 8 = 16*

*Difference in ratio of B and C out of 16 = 8 – 5 = 3*

*Total amount = Rs. 3200*

*∴ Difference of shares between B and C = Rs. ^{3}⁄_{16} × 3200 *

* = Rs. 3 × 200*

* = Rs. 600 ……………..(Ans.)*

**Q.6)** *The average of 5 members of a family is 24 years. If the youngest member is 8 years old, then what was the average age (in years) of the family at the time of the birth of the youngest member ?*

a) 16

b) 20

c) 24

d) 32

## Answer

Correct Answer:** (b)** 20

## Solution

*The average age of 5 members = 5 × 24 = 120 years*

*The age of the youngest member = 8 years.*

*But, as per the statement of the question given, time of the birth of the youngest member means 8 years ago.*

*Hence, before 8 years their (4 members) total ages = 120 – 5 × 8 = 80 years.*

*∴ Average age of the family = ^{80}⁄_{4}*

*years*

* **=* *20 years. ……………..( Ans.)*

**Q.7)** *If the price of pen decreases by 20%, then a man can buy 10 more pens for Rs 100. What is the new price (in Rs) of each pen ?*

a) 1

b) 2

c) 3

d) 4

## Answer

Correct Answer:** (b)** 2

## Solution

*Let, initial price of each pen = x and total no. of pen (in Rs.100) = y*

*∴ Total price of x no. of pens = xy*

*Given, xy = Rs. 100*

*Hence, y = ^{100}⁄_{x}*

*After decreasing the price of pen, new price of each pen,*

* = Rs.x × ^{(100 − 20)}⁄_{100} *

* = Rs. ^{4x}⁄_{5} *

*Number of pen that he can buy in Rs. 100, we have *

* = 100 ÷ ^{4x}⁄_{5}*

* = ^{125}⁄_{x}*

*Now, we have,*

⇒ ^{125}⁄_{x} − y = 10

⇒ ^{125}⁄_{x} − ^{100}⁄_{x} = 10

*∴ x = Rs. 2.5*

*∴ New price = Rs. ^{4x}⁄_{5} *

* = Rs. ^{(4 × 2.5)}⁄_{5}*

* = Rs. 2 ……………………(Ans.)*

**Q.8)** *After deducting 60% from a certain number and then deducting 15% from the remainder, 1428 is left. What was the initial number ?*

a) 4200

b) 3962

c) 4150

d) 4300

## Answer

Correct Answer:** (a)** 4200

## Solution

*Let, the actual number be y*

*∴ After two successive deduction of 60 % and 15 %, the number we have left 1428.*

*∴ We can write,*

*⇒ ^{(100 − 60)}⁄_{100} × ^{(100 − 15)}⁄_{100} × y = 1428 *

*∴ y = 4200 ……………………(Ans.)*

**Q.9)** *A train travels 40% faster than a car. Both start from point A at the same time and reach point B, 140 km away at the same time. On the way the train takes 25 minutes for stopping at the stations. What is the speed (in km/hr) of the train ?*

a) 67

b) 134.4

c) 145.9

d) 160

## Answer

Correct Answer:** (b)** 134.4

## Solution

*Let, speed of the car = x km/hr*

*∴ Speed of the train = x × ^{(100 + 40)}⁄_{100}*

*km/hr*

* = ^{7x}⁄_{5}*

*km/hr*

*Now, according to the question given, we have*

*T _{car} – T_{train} = 25 minutes*

⇒ 140 V car − 140 V train = 25 60 hours ⇒ 140 x − 140 ( 7x 5 ) = 25 60 ⇒ 140 x ( 1− 5 7 )= 25 60 ⇒x= 60 25 × 2 7 ×140

*∴ x = 96 hour*

*Hence, Speed of the train = ^{7x}⁄_{5}*

*km/hr*

* = ^{(7 × 96)}⁄_{5} = 134.4 km/hr ……………..(Ans.)*

**Q.10)** *A certain sum of money triples itself in 5 years at simple interest. In how many years it will be five times ?*

a) 5

b) 8

c) 10

d) 15

## Answer

Correct Answer:** (c)** 10

## Solution

*Let, principle (P) = x*

*∴ Amount (A) = 3x*

*Hence, Interest (I) = 3x − x = 2x*

*Time (t) = 5 years.*

*∴ Rate of interest (r) = ^{(100 × I)}⁄_{(P × t)}*

* r = *^{(100 × 2x)}⁄_{(x × 5) = 40 %}

*Now, again, A = 5x*

*∴ I = 5x − x = 4x*

*r = 40 %*

*t = ?*

*∴ t = ^{(100 × I)}⁄_{(P × r)}*

* t = *^{(100 × 4x)}⁄_{(x × 40)}* = 10 years. ……………………….(Ans.)*

**Q.11)** *If *

x+1x=2

, then what is the value of x^{64} + x^{121} ?

a) 0

b) 1

c) 2

d) – 2

## Answer

Correct Answer:** (c)** 2

## Solution

*If x + ^{1}⁄_{x} = 2 , then x = 1*

*∴ x ^{64} + x^{121} = (1)^{64} + (1)^{121} = 1 + 1 = 2 ………………..(Ans.)*

**Q.12)** *If *

x=6+26

, then what is the value of

x-1+1x-1

?

a) 2√3

b) 3√2

c) 2√2

d) 3√3

## Answer

Correct Answer:** (a)** 2√3

## Solution

*x = 6 + 2√6*

*∴ **x − 1 = 5 + 2√6*

* = (√3) ^{2} + (√2)^{2} + 2 x √3 x √2*

* = (√3 + √2) ^{2}*

*Hence, *√(x − 1)*= (√3 + √2) (taking +ve value, since x is greater than 1)*

*As, we have seen that (√3) ^{2} − (√2)^{2} = 1*

*∴ *The value of ^{1}⁄_{√(x − 1)}* should be = **(√3 − √2)*

*∴ *√(x − 1) + ^{1}⁄_{√(x − 1)} = (√3 + √2) + (√3 − √2) = 2√3* ……………………( Ans.)*

**Q.13)** *If a + b + c = 27, then what is the value of (a – 7) ^{3} + (b – 9)^{3} + (c – 11)^{3} – 3(a – 7)(b – 9)(c – 11) ?*

a) 0

b) 9

c) 27

d) 81

## Answer

Correct Answer:** (a)** 0

## Solution

*Let, (a – 7) = x, (b – 9) = y and (c – 11) = z*

*∴ **x + y + z = (a – 7) + (b – 9) + (c – 11) = (a + b + c) – 27 = 27 – 27 = 0*

*We know that if x + y + z = 0, then*

*x ^{3} + y^{3} + z^{3} – 3xyz = 0*

*∴** (a – 7)*^{3}* **+ (b **–** 9)*^{3}* **+ (c **–** 11)*^{3}* **–** 3(a **–** 7)(b **–** 9)(c **–** 11) = 0 ……….(Ans.)*

**Q.14)** *If *

x=6+26

, then what is the value of

x+5x-5+x+3x-3

?

a) √5

b) √3

c) √15

d) 2

## Answer

Correct Answer:** (d)** 2

## Solution

*Note:**If the value of x has given in such way that,*

*x = ^{2ab}⁄_{(a + b)}*

*and the required question has asked in such way that*

^{(x + a)}⁄_{(x − a)}+^{(x + b)}⁄_{(x − b)}= ?*Then the required value of ^{(x + a)}⁄_{(x − a)} + ^{(x + b)}⁄_{(x − b)} should be = 2*

*In this problem, if you assume a = √5 and b = √3, then we can write the sum like that*

*x = ^{2√15}⁄_{(√5 + √3)} = ^{2√5 × √3}⁄_{(√5 + √3)}*

*∴ ^{(x + √5)}⁄_{(x − √5)} + ^{(x + √3)}⁄_{(x − √3)} = 2 *

*…………..(*

**Ans.**)

__Detailed Process:__

x= 2 15 5 + 3 ⇒x= 2 5 ⋅ 3 5 + 3 ⇒ x 5 = 2 3 5 + 3 ⇒ x+ 5 x− 5 = 2 3 +( 5 + 3 ) 2 3 −( 5 + 3 ) ⇒ x+ 5 x− 5 = 2 3 + 5 3 − 5

*Similarly, *

x+ 3 x− 3 = 3 5 + 3 5 − 3

*Now L.H.S.,*

x+ 5 x− 5 + x+ 3 x− 3 ⇒ 3 3 + 5 3 − 5 + 3 5 + 3 5 − 3 ⇒ 3 3 + 5 −3 5 − 3 ( 3 − 5 ) ⇒ 2 3 −2 5 ( 3 − 5 ) ⇒ 2( 3 − 5 ) ( 3 − 5 ) =2

**Q.15)** *The perimeter of an isosceles triangle is 32 cm and each of the equal sides is 5/6 times of the base. What is the area (in cm ^{2}) of the triangle ?*

a) 39

b) 48

c) 57

d) 64

## Answer

Correct Answer:** (b)** 48

## Solution

*Let, Length of base = b*

*∴** Length of each equal side (a) = ^{5b}⁄_{6}*

*∴** We have,*

*Perimeter = 32 cm*

*∴** 2a + b = 32*

*or, 2 × ^{5b}⁄_{6} + b = 32*

*∴** b = 12 cm and *

* a = ^{5b}⁄_{6} = ^{(5 × 12)}⁄_{6} = 10 cm.*

*Height (h) = √[a ^{2} − ( ^{b}⁄_{2} )^{2}] *

* = √[10 ^{2} − ( ^{12}⁄_{2} )^{2}] *

* = √[100− 36]*

* = 8 cm.*

*We know that, area of an isosceles triangle,*

* = **½ × b × h** sq.cm. *

*= ½ x 12 x 8** sq.cm.*

*= 48 sq.cm …………………(Ans.)*

**Q.16)** *If length of each side of a rhombus PQRS is 8 cm and ∠PQR = 120°, then what is the length (in cm) of QS ?*

a) 4√5

b) 6

c) 8

d) 12

## Answer

Correct Answer:** (c)** 8

## Solution

*As we know that diagonal of a rhombus bisects its angle.*

*Here, ∠ PQR = 120°*

*Line QS is one diagonal which bisects ∠ PQR.*

*∴** ∠ PQS = 60°*

*As, PQ = PS = 8 cm.*

*Therefore, ∠ PSQ = 60°*

*Hence, remaining ∠ QPS = 60°*

*∴** ∆ PQS is an equilateral triangle.*

*∴** The required length of side QS = 8 cm. *

*……………………….(Ans.)*

**Q.17)** *In the given figure, ABC is a triangle. The bisectors of internal ∠ B and external ∠ C intersects at D. If ∠ BDC = 48°, then what is the value (in degrees) of ∠ A ?*

a) 48

b) 96

c) 100

d) 114

## Answer

Correct Answer:** (b)** 96

## Solution

*In **∆** BCD, external **∠** DCE = internal (**∠** CBD + **∠** BDC)*

* **∴** x = 48° + y*

*Again, In ∆ BCD,*

*External **∠** ACE = internal (**∠** CBA + **∠** A)*

*∴** x + x = y + y + **∠** A*

*or, 2x = 2y + **∠** A*

*or, **∠** A = 2y – 2x *

*or, **∠** A = 2y – 2(48° + y)*

*∴ ∠ A = 96° …………………..(Ans.)*

**Q.18)** *In the given figure, O is the centre of the circle and ∠ DCE = 45°. If CB = BD and CD = 10√2 cm, then what is the length (in cm) of AC ?*

a) 14

b) 15.5

c) 18.5

d) 20

## Answer

Correct Answer:** (c)** 18.5

## Solution

**Note:** The length of the AC can’t be determined if it is not state that *∠** OCE = 90°*

*Now, we have **∠** DCE = 45°*

*∴** **∠** OCB = 45°*

*Hence, we can say ∆OBC is a right-angled isosceles triangle means BC = OB = 10√2/2 cm = 5√2 cm.*

*In right-angled isosceles triangle OBC, we have, *

*OC ^{2} = BO^{2} + BC^{2 } [using ‘Pythagoras theorem’]*

*⇒** OC ^{2} = (5√2)^{2} + (5√2)^{2}*

*∴** OC ^{2} = 100 *

*⇒*

*OC = 10 cm.*

*∴** OC = AB = 10 [ OC = AB, radius of same circle ]*

*Now, In right-angled ∆ ABC, we have*

*AC ^{2} = BC^{2} + AB^{2}*

*⇒** AC ^{2} = (5√2)^{2} + (OB + AO)^{2}*

*⇒** AC ^{2} = (5√2)^{2} + (5√2 + 10)^{2}*

*⇒** AC ^{2} = 50 + 50 + 2 x 5√2 x 10 + 100*

*∴ AC = √341.42 = 18.5 cm. (approx.) ………………….(Ans.)*

**Q.19)** *What is the simplified value of *

sin2A1+cos2A

?

a) *tan A*

b) *cot A*

c) *sin A*

d) *cos A*

## Answer

Correct Answer:** (a)** *tan A*

## Solution

*We know that 1 + cos 2A = 2cos ^{2 }A and sin2 A = 2sin A.cos A*

*∴ ^{sin 2A}⁄_{(1 +cos 2A)} = ^{2 sin A.cos A}⁄_{2 cos2 A} = tanA ……………..(Ans.)*

**Q.20)** *What is the simplified value of *

secAtanA+cotA2

?

a) *1 – cos ^{2}A*

b) *2sin ^{2}A*

c) *sec ^{2}A*

d) *cosec ^{2}A*

## Answer

Correct Answer:** (a)** *1 – cos ^{2}A*

## Solution

*Let A = 45°*

*Then sec A = sec 45° = √2*

*tan A + cot A = tan 45° + cot 45° = 1 + 1 = 2*

*Now *

( secA tanA+cotA ) 2 ⇒ ( 2 2 ) 2 = 1 2 ⇒1− cos 2 A=1− cos 2 45 o = 1 2

**Q.21)** *What is the simplified value of 1 + tan A tan (A/2) ?*

a) *sin A/2*

b) *cos A*

c) *sec A*

d) *sin A*

## Answer

Correct Answer:** (c)** *sec A*

## Solution

*Let A = 60°*

*∴ 1 + tan A × tan ( ^{A}⁄_{2} ) = 1 + tan 60° × tan 30°*

*⇒ 1 + √3 × ^{1}⁄_{√3} = 2 = sec^{2} A = sec^{2} 60° = 2*

**Q.22)** *The ‘Pie Chart’ given below shows the number of shoes of five different brands in a multi brand store. There are total 1200 shoes. *

How many shoes are there of Reebok brand ?

a) 230

b) 276

c) 286

d) 216

## Answer

Correct Answer:** (b)** 276

## Solution

*Total number of shoes = 1200 on which Reebok shoe = 23 %*

*∴ Number of Reebok shoes = 1200 × ^{23}⁄_{100} = 276 ……………(Ans.)*

**Q.23)** *The ‘Pie Chart’ given below shows the number of shoes of five different brands in a multi brand store. There are total 1200 shoes. *

What is the difference in number of shoes of Puma and Vans ?

a) 96

b) 156

c) 84

d) 112

## Answer

Correct Answer:** (a)** 96

## Solution

*Total number of shoes = 1200*

*Percentage of Puma shoe = 21 % and Vans = 13 %*

*Difference in percentage = 21 – 13 = 8 %*

*∴ Difference in number of Puma and Vans shoes = 1200 × ^{8}⁄_{100} = 96 ………(Ans.)*

**Q.24)** *The ‘Pie Chart’ given below shows the number of shoes of five different brands in a multi brand store. There are total 1200 shoes. *

The difference between the number of shoes of Reebok and Nike is same as the difference between which of the following two brands ?

a) Puma and Adidas

b) Reebok and Adidas

c) Vans and Nike

d) Nike and Adidas

## Answer

Correct Answer:** (c)** Vans and Nike

## Solution

*From the adjoining figure, we have clearly seen that the difference in the percentage of Reebok and Nike is 5 % and we also see that the difference in percentage is same means 5 % for Nike and Vans.*

*∴ Answer will be Nike and Vans. ……………………..(Ans.)*

**Q.25)**

Puma shoes are how much percent more than the Nike Shoes ?

a) 14.28

b) 16.66

c) 25

d) 21.33

## Answer

Correct Answer:** (b)** 16.66

## Solution

*The number of Nike shoes = 1200 × ^{18}⁄_{100} = 216*

*The number of Puma shoes = 1200 × ^{21}⁄_{100} = 252*

*∴ The number of excess shoes of Puma than Nike = 252 – 216 = 36*

*∴ in percentage = ^{36}⁄_{216} × 100 % = 16.66 % (approx.) ……………….(Ans.)*


To construct a figure similar to a given rectilineal figure and equal to another given rectilineal figure, let's consider the given figures. Let ABC be the rectilineal figure to which the constructed figure must be similar, and D be the figure to which it must be equal. The goal is to construct a figure that is both similar to ABC and equal to D.

First, a parallelogram BE is applied to BC, equal to the triangle ABC. Then, a parallelogram CM is applied to CE, equal to D, in an angle equal to the angle CBL. As a result, BC is in a straight line with CF, and LE is in a straight line with EM.

Next, a mean proportional GH is taken to BC and CF. On GH, a figure KGH is described, similar and similarly situated to ABC. Since BC is to GH as GH is to CF, and given the properties of proportional straight lines, the triangle ABC is to the triangle KGH as BC is to CF. Additionally, the parallelogram BE is to the parallelogram EF as BC is to CF.

Therefore, the triangle ABC is to the triangle KGH as the parallelogram BE is to the parallelogram EF. By alternation, the triangle ABC is to the parallelogram BE as the triangle KGH is to the parallelogram EF. Since the triangle ABC is equal to the parallelogram BE, the triangle KGH is equal to the parallelogram EF. Given that the parallelogram EF is equal to D, KGH is also equal to D. Moreover, KGH is similar to ABC, fulfilling the requirement of being similar to ABC and equal to D. Thus, the constructed figure KGH meets the specified conditions.


**Grade 3 Equivalence Lessons**

The provided lessons and exercises aim to help Grade 3 students understand equivalence by manipulating units and reasoning about their size, aligned with Common Core Standards 3.NF.3a, 3.NF.3b, and 3.NF.3c.

**Exercises**

1. **Equivalent Fractions**: Use a tape diagram to model equivalent fractions.
   - 2 tenths is equal to _____ fifths.
   - 1 third is equal to _____ ninths.
   - Observe how the size of equal parts changes when there are fewer equal parts.

2. **Sharing Pizzas**: 8 students want to share 2 identical pizzas, one cut into 4 slices and the other into 8 slices. How can they share the pizzas equally without breaking any pieces?

3. **Comparing Fractions**:
   - Why does it take 4 copies of 1 tenth to show 2 copies of 1 fifth? Draw a model to support your answer.
   - How many eighths does it take to make 1 fourth? Draw a model to support your answer.

4. **Cake Slices**: Mr. Pham cuts a cake into 8 equal slices and then cuts each slice in half. How many small slices does he have? Explain using words and numbers.

**Application Problem**

A 2-meter branch is cut into 1/6 meter pieces.
- Draw a number line to show the total length of the branch, partitioning and labeling each cut.
- How many pieces are there altogether?
- Write 2 equivalent fractions to describe the total length of the branch.

**Concept Development**

Divide a rectangle into thirds. To double the units, cut each third in half, resulting in sixths. Repeat this process to triple the units. Help students understand equivalence by comparing different units and sizes, and how the more a whole is cut, the smaller the pieces become.

**Homework**

Repeat exercises 1 through 5, focusing on understanding equivalent fractions, sharing equal parts, comparing fractions, and explaining the relationship between the number of cuts and the size of the pieces.


The sum of the internal angles of a triangle on a sphere is greater than 180° and is related to the area of the triangle. The area of a spherical triangle with interior angles α, β, γ is given by the formula S = r^{2} (α + β + γ - π), which is notable for not containing the lengths of the sides.

This formula can be derived using the concept of a diangle, which is the figure enclosed by two great circles on a sphere. The area of a diangle is given by S = 2r^{2}θ, where θ is the angle of the diangle. By extending each side of a spherical triangle to form great circles, six diangles are created, which cover the entire sphere. The sum of the areas of these diangles is equal to the area of the sphere plus four times the area of the triangle.

The area of a spherical triangle can also be expressed in terms of the angle turned through when walking around the triangle. When walking around a triangle on a sphere, the total angle turned through is Δ = 3π - α - β - γ. The area of the triangle can be expressed as S = r^{2}(2π - Δ), which shows that the area is related to the amount of turning required to circumambulate the triangle.

For a triangle in a plane, Δ is always 2π radians, resulting in S = 0. However, on a sphere, Δ can be less than 2π, and the area of the triangle is non-zero. In the case where the path is a great circle, Δ = 0, and the area enclosed is a hemisphere. The formula for the area of a spherical triangle can be generalized to S = r^{2}(2π - ∫ kds), where k is the geodesic curvature of the path.

This formula, known as the Gauss-Bonnet formula, relates the area of a spherical triangle to the curvature of the path that encloses it. The geodesic curvature k is zero for a great circle, and the formula can be used to calculate the area of a spherical triangle given the curvature of the path and the internal angles of the triangle.


The first five attributes are Boolean, and the last is an integer "rating." Points with smaller angles are more similar. Compute a symmetric matrix of distances (or similarities) between the rows or columns of a matrix; or compute cross-distances between the rows or columns of two different matrices. We now create two vectors: x . Cosine similarity is a measure of distance between two vectors. Therefore it is my understanding that by normalising my original dataset through the code below. Points with larger angles are more different. We will show you how to calculate the Euclidean distance and construct a distance matrix. Pay attention to this fact; if you forget, the resulting bugs may bite you hard in the, er, leg. Cosine Similarity using R - Comparison with Euclidean Distance Anyway, this is why the typical ‘distance’ algorithm like ‘Euclidean’ won’t work well to calculate the similarity. minkowski: The p norm, the pth root of the sum of the pth powers of the differences of the components. In our example the angle between x14 and x4 was larger than those of the other vectors, even though they were further away. Intuitively, let’s say we have 2 vectors, each representing a sentence. and also, Scikit-learn's distance metrics doesn't have cosine distance. You just divide the dot product by the magnitude of the two vectors. It can be proven by dividing the triangle into two right ones and using the above definition of sine. This similarity measure is typically expressed by a distance measure such as the Euclidean distance, cosine similarity or the Manhattan distance. First the Theory. Search the textTinyR package. Namely, magnitude. Instead, use a special variable called pi. This tutorial explains how to calculate the Cosine Similarity between vectors in Python using functions from the NumPy library.. Cosine Similarity Between Two Vectors in Python Cosine distance. This series is part of our pre-bootcamp course work for our data science bootcamp. It is a symmetrical algorithm, which means that the result from computing the similarity of Item A to Item B is the same as computing the similarity of Item B to Item A. Here is the code for LSH based on cosine distance: from __future__ import division import numpy as np import math def signature_bit(data, planes): """ LSH signature generation using random projection Returns the signature bits for two data points. In NLP, this might help us still detect that a much longer document has the same “theme” as a much shorter document since we don’t worry about the magnitude or the “length” of the documents themselves. Distance Based Metrics: Euclidean distance; Manhattan distance; Similarity Based Metrics . A class Cosine defined two member functions named "similarity" with parameter type difference, in order to support parameters type int and double 2-D vectors. In other words, the similarity to the data that was already in the system is calculated for any new data point that you input into the system. November 24, 2014 Leave a comment. Both class (static) member function similarity can be invoked with two array parameters, which represents the vectors to measure similarity between them. textTinyR Text Processing for Small or Big Data Files. Cosine distance includes a dot product scaled by norms: Cosine distance includes a dot product scaled by Euclidean distances from the origin: CosineDistance of vectors shifted by their means is equivalent to CorrelationDistance: See Also. Missing values are allowed, and are excluded from all computations involving the rows within which they occur. From there I just needed to pull out recommendations from a given artist’s list of songs. Cosine similarity is the cosine of the angle between 2 points in a multidimensional space. It is also not a proper distance in that the Schwartz inequality does not hold. … Toggle navigation Brad Stieber. Because cosine distances are scaled from 0 to 1 (see the Cosine Similarity and Cosine Distance section for an explanation of why this is the case), we can tell not only what the closest samples are, but how close they are. Curse of dimensionality) Calculate Cosine Similarity with Exploratory. Data, R code and supplemental material. I came across this calculation when I was reading about Recommender systems. Cosine distance; Euclidean distance; Relaxed Word Mover’s Distance; Practical examples. Cosine Similarity is a measure of the similarity between two vectors of an inner product space.. For two vectors, A and B, the Cosine Similarity is calculated as: Cosine Similarity = ΣA i B i / (√ΣA i 2 √ΣB i 2). where R is the triangle's circumradius. The cosine similarity is advantageous because even if the two similar documents are far apart by the Euclidean distance because of the size (like, the word ‘cricket’ appeared 50 times in one document and 10 times in another) they could still have a smaller angle between them. This code doesn’t give you the correct result, however, because R always works with angles in radians, not in degrees. Then, I’ll look at the math behind cosine similarity. The last column is the rating given by a particular user for a movie. In this post, we will be looking at a method named Cosine Similarity for item-based collaborative filtering. While cosine looks at the angle between vectors (thus not taking into regard their weight or magnitude), euclidean distance is similar to using a ruler to actually measure the distance. It would be good to have a better name for the weird metric. I am currently solving a problem where I have to use Cosine distance as the similarity measure for k-means clustering. The content we watch on Netflix, the products we purchase on Amazon, and even the homes we buy are all served up using these algorithms. A distance matrix in the form of an object of class dist, of the sort returned by the dist function or the as.dist function. However, the following angular definitions are proper distances: Pearson’s Correlation. Examples Author(s) Kevin R. Coombes See Also. Cosine similarity is not a distance metric as it violates triangle inequality, and doesn’t work on negative data. However, cosine similarity is fast, simple, and gets slightly better accuracy than other distance metrics on some datasets. While harder to wrap your head around, cosine similarity solves some problems with Euclidean distance. Transcript . $\endgroup$ – Smith Volka Sep 5 '17 at 8:16. I will not go into depth on what cosine similarity is as the web abounds in that kind of content. The signature bits of the two points are different only for the plane that divides the two points. """ Smaller the angle, higher the similarity. Though the notion of the cosine was not yet developed in his time, Euclid's Elements, dating back to the 3rd century BC, contains an early geometric theorem almost equivalent to the law of cosines.The cases of obtuse triangles and acute triangles (corresponding to the two cases of negative or positive cosine) are treated separately, in Propositions 12 and 13 of Book 2. 6 Only one of the closest five texts has a cosine distance less than 0.5, which means most of them aren’t that close to Boyle’s text. If I am using cosine similarity, would it be the highest cosine similarity? So, you may want to try to calculate the cosine of an angle of 120 degrees like this: > cos(120) [1] 0.814181. Articles Related Formula By taking the algebraic and geometric definition of the Then, you use this similarity value to perform predictive modeling. From Wikipedia: “Cosine similarity is a measure of similarity between two non-zero vectors of an inner product space that “measures the cosine of the angle between them” C osine Similarity tends to determine how similar two words or sentence are, It can be used for Sentiment Analysis, Text Comparison and being used by lot of popular packages out there like word2vec. Instead, we want to use the cosine similarity algorithm to measure the similarity in such a high-dimensional space. CorrelationDistance EuclideanDistance. Cosine similarity works in these usecases because we ignore magnitude and focus solely on orientation. BUGS. cosine distance of two character strings (each string consists of more than one words) rdrr.io Find an R package R language docs Run R in your browser R Notebooks. Description: Euclidian Distance vs Cosine Similarity for Recommendations. Here’s how to do it. In wordspace: Distributional Semantic Models in R. Description Usage Arguments Value Distance Measures Author(s) See Also Examples. As usual we will use built-in text2vec::moview_review dataset. 1 $\begingroup$ You can simply convert the distance into similarity. Recommendation engines have a huge impact on our online lives. Cosine similarity; Jaccard similarity; 2. For this reason, a vast portfolio of time series distance measures has been published in the past few years. However, the standard k-means clustering package (from Sklearn package) uses Euclidean distance as standard, and does not allow you to change this. Vignettes. ... (R) and Bradley (B) have rated the movies. The Cosine Similarity procedure computes similarity between all pairs of items. If you want the magnitude, compute the Euclidean distance instead. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. We don’t compute the similarity of items to themselves. However, to find the most nearest points to the centroid he uses the minimum cosine distance. Euclidean distance and cosine similarity are the next aspect of similarity and dissimilarity we will discuss. Tutorials Partitioning Data into Clusters; Related Guides Distance and Similarity Measures; History. While there are libraries in Python and R that will calculate it sometimes I’m doing a small scale project and so I use Excel. So when we’ve got real values– and this is sort of a primer for the boot. Similarity based methods determine the most similar objects with the highest values as it implies they live in closer neighborhoods. The cosine similarity is a measure of the angle between two vectors, normalized by magnitude. Distance Measures for Time Series in R: The TSdist Package by Usue Mori, Alexander Mendiburu and Jose A. Lozano Abstract The deﬁnition of a distance measure between time series is crucial for many time series data mining tasks, such as clustering and classiﬁcation. We can therefore compute the score for each pair of nodes once. The distance is the proportion of bits in which only one is on amongst those in which at least one is on. dist, as.dist. Complete Series: Introduction to Text Analytics in R. More Data Science Material: [Video Series] Beginning R Programming [Video] Euclidean Distance & Cosine Similarity – Data Mining Fundamentals Part 18 [Blog] Feature Engineering and Data Wrangling in R (2108) In this tutorial, we will introduce how to calculate the cosine distance between two vectors using numpy, you can refer to our example to learn how to do. If distance from A to B is 0.3, then the similarity will be 1-0.3=0.7. WEIGHTED COSINE DISTANCE WEIGHTED COSINE SIMILARITY Name: WEIGHTED CORRELATION (LET) WEIGHTED COVARIANCE (LET) WEIGHTED COSINE DISTANCE (LET) WEIGHTED COSINE SIMILARITY (LET) Type: Let Subcommand Purpose: Compute the weighted correlation coefficient between two variables. 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The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity centroid he uses the minimum cosine distance as the web abounds in that kind content! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! The distance into similarity distance from a to B is 0.3, then the similarity will be 1-0.3=0.7 the! 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**STATS 7059 Mathematical Statistics**
**North Terrace Campus, Semester 1, 2022**

**General Course Information**

* Course Code: STATS 7059
* Course Name: Mathematical Statistics
* Coordinating Unit: Mathematical Sciences
* Term: Semester 1
* Level: Postgraduate Coursework
* Location: North Terrace Campus
* Units: 3
* Contact: Up to 3 hours per week
* Available for Study Abroad and Exchange: Yes
* Assumed Knowledge: (MATHS 1012 and STATS 2107) or (MATHS 2201 and MATHS 2202 and STATS 2107) or (MATHS 2106 and MATHS 2107 and STATS 2107)

**Course Description**
Statistical methods used in practice are based on a foundation of statistical theory. This course is concerned with the fundamental theory of random variables and statistical inference. Topics covered include:
* Calculus of distributions, moments, moment generating functions
* Multivariate distributions, marginal and conditional distributions, conditional expectation and variance operators
* Weak convergence, convergence in distribution, weak law of large numbers, central limit theorem
* Statistical inference, likelihood, score and information
* Estimation, minimum variance unbiased estimation, the Cramer-Rao lower bound, exponential families, sufficient statistics
* Tests of hypotheses, most powerful tests, the Neyman-Pearson lemma, likelihood ratio, score and Wald tests

**Course Staff**
* Course Coordinator: Dr Melissa Humphries

**Learning Outcomes**
On successful completion of this course, students will be able to:
1. Demonstrate knowledge of, and properties of, statistical models in common use
2. Understand the basic principles underlying statistical inference (estimation and hypothesis testing)
3. Construct tests and estimators, and derive their properties
4. Demonstrate knowledge of applicable large sample theory of estimators and tests

**University Graduate Attributes**
This course will provide students with an opportunity to develop the following Graduate Attributes:
* Deep discipline knowledge and intellectual breadth
* Creative and critical thinking, and problem-solving
* Teamwork and communication skills
* Professionalism and leadership readiness
* Self-awareness and emotional intelligence

**Learning Resources**
* Required Resources: A set of lecture notes will be provided
* Recommended Resources:
	+ Mathematical Statistics with Applications (7th ed.), by D.D. Wackerly, W. Mendenhall, and R.L. Scheaffer, Duxbury Press
	+ Mathematical Statistics and Data Analysis (3rd ed.), by J.A. Rice, Duxbury Press
	+ Statistical Inference (2nd ed.), by G. Casella and R. L. Berger, Duxbury Press
	+ Modern Mathematical Statistics with Applications (2nd ed.), by J.L. Devore and K.N. Berk, Springer

**Learning & Teaching Activities**
* Lectures: 30 hours
* Tutorials: 5 hours
* Assessments: 14 hours
* Total: 156 hours

**Assessment**
* Assignments: 30% (3 assignments, each 10% of final grade)
* Quizzes: 10% (10 quizzes, each 1% of final grade)
* Mid-Semester Test: 30% of final grade
* End-of-Semester Test: 30% of final grade
* An aggregate score of at least 50% is required to pass the course

**Assessment Detail**
* Five equally weighted (10% each) assignments, due at the end of weeks 3, 5, 7, 9, 12
* The assignments will be distributed on Monday of weeks 2, 4, 6, 8, 11

**Submission**
* All written assignments are to be submitted online via MyUni
* Late assignments will not be accepted unless an extension has been arranged prior to the due date

**Course Grading**
Grades for your performance in this course will be awarded in accordance with the following scheme:
* FNS: Fail No Submission
* F: 0-49%
* P: 50-64%
* C: 65-74%
* D: 75-84%
* HD: 85-100%
* CN: Continuing
* NFE: No Formal Examination
* RP: Result Pending

**Student Feedback**
The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways, including ongoing engagement with staff, the use of online discussion boards, and the use of Student Experience of Learning and Teaching (SELT) surveys.

**Student Support**
* Academic Integrity for Students
* Academic Support with Maths
* Academic Support with writing and study skills
* Careers Services
* International Student Support
* Library Services for Students
* LinkedIn Learning
* Student Life Counselling Support
* Students with a Disability: Alternative academic arrangements
* YouX Student Care: Advocacy, confidential counselling, welfare support and advice

**Policies & Guidelines**
This section contains links to relevant assessment-related policies and guidelines, including:
* Academic Credit Arrangements Policy
* Academic Integrity Policy
* Academic Progress by Coursework Students Policy
* Assessment for Coursework Programs Policy
* Copyright Compliance Policy
* Coursework Academic Programs Policy
* Elder Conservatorium of Music Noise Management Plan
* Intellectual Property Policy
* IT Acceptable Use and Security Policy
* Modified Arrangements for Coursework Assessment Policy
* Reasonable Adjustments to Learning, Teaching & Assessment for Students with a Disability Policy
* Student Experience of Learning and Teaching Policy
* Student Grievance Resolution Process

**Fraud Awareness**
Students are reminded that the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment.


## Square Root of a Number

The square root of a number is a value that, when multiplied by itself, equals the given number. It is denoted by √x, where x is the given number.

## Important Concepts for Square Root and Cube Root

### Square Root of Factorization
When a given number is a perfect square, it can be resolved into prime factors, and the product of these prime factors can be taken, choosing one out of every pair.

### Examples

**Example 1:** The square root of a number x is denoted by √x.

**Example 2:** Evaluate √(4+3.8729).
Solution: √(4+3.8729) = √7.8729.

**Example 3:** If √37 = 6.08 (approximately), find the value of 6.08^2.
Solution: Given expression is (6.08)^2 = 36.9664 ≈ 37.

## Cube Root of a Number

The cube root of a number x is the number whose cube is x, denoted by ∛x.

### Cube Root of Factorization
Resolve the given number into prime factors and take the product of the prime factors.

### Example
Thus, ∛343 = ∛(7^3) = 7.

## Key Concepts for Square and Cube Roots

- Square root: √x, where x is the given number.
- Cube root: ∛x, where x is the given number.
- Factorization for square and cube roots involves resolving the number into prime factors and taking the product of these factors accordingly.


To analyze the World Series results from 1903 to the present, we start with a simplified model that ignores home-field advantage, assuming each team has a 0.5 probability of winning each game. This model gives expected probabilities for a best-of-seven series: P(4) = 0.125, P(5) = 0.250, P(6) = 0.3125, and P(7) = 0.3125.

After cleaning the data by removing best-of-nine series and those with ties, we are left with 95 series. Comparing actual and expected probabilities:

- 4-0: Expected 0.125, Actual 0.179
- 4-1: Expected 0.250, Actual 0.221
- 4-2: Expected 0.3125, Actual 0.242
- 4-3: Expected 0.3125, Actual 0.358

The World Series has been swept 17 times, which is more than the model predicts. Using the BINOMDIST function, we find that if the model's probabilities were correct, 17 or more sweeps in 95 occurrences would happen only 8% of the time, just shy of statistical significance.

Further analysis reveals that the series went to seven games 34 out of 95 times, which is more than the model predicts. This would occur randomly only 20% of the time. Focusing on the team behind 3-2 winning the sixth game, we find this happened 34 out of 57 times, a somewhat unusual result.

To achieve statistical significance, we narrow our data set to 1945-2002. In these 58 years, the World Series lasted seven games 27 times, which would happen by chance only 1% of the time. The team behind 3-2 won the sixth game 27 of 39 times, again a 1% chance.

Possible explanations for these anomalies include the team behind playing harder or the leading team choking. However, testing these explanations against NBA Finals data shows the opposite trend, suggesting that the baseball results might be due to random variation.

Further commentary suggests that teams might throw in the towel after falling behind 3-0, as historically, no team had come back from such a deficit until the 2004 Red Sox. The simple model predicts a team should rally from a 3-0 deficit once in 16 tries, but this has happened only three times out of 20.

The Chi-squared test can be used to determine if the observed frequencies significantly differ from the expected frequencies. Calculating the sum of (Ai-Ei)^2/Ei gives 4.68, which is a chi-squared variable with 3 degrees of freedom. The cutoff for p=0.05 significance is 7.82, meaning we cannot reject the Null hypothesis that the distribution of results is generated by independent coin flips.

In conclusion, while there are intriguing anomalies in World Series data, such as the frequency of sweeps and the tendency for series to go to seven games, these may not be statistically significant when properly tested. Theories about team behavior, such as giving up or choking, are plausible but not supported by data from other sports. The World Series, named for its historical sponsorship by the newspaper the World, is a best-of-seven series that has been played every year but two since 1903, with participation currently limited to North American teams.