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The Central Limit Theorem for Sumsz-score and standard deviation for sums:zfor the sample mean =Σxâ(n)(μX)(n)(ÏX)zfor the sample mean =Σxâ(n)(μX)(n)(ÏX) | https://openstax.org/books/introductory-statistics-2e/pages/7-formula-review |
Standard deviation for Sums (âX):(n)(n)(Ïx) | https://openstax.org/books/introductory-statistics-2e/pages/7-formula-review |
Average : a number that describes the central tendency of the data; there are a number of specialized averages, including the arithmetic mean, weighted mean, median, mode, and geometric mean. | https://openstax.org/books/introductory-statistics-2e/pages/7-key-terms |
Central Limit Theorem : Given a random variable (RV) with known meanμand known standard deviation,Ï, we are sampling with sizen, and we are interested in two new RVs: the sample mean,X¯X¯, and the sample sum,ΣΧ. If the size (n) of the sample is sufficiently large, thenX¯X¯~N(μ,ÏnÏn) andΣΧ~N(nμ, (nn)(Ï)).... | https://openstax.org/books/introductory-statistics-2e/pages/7-key-terms |
Exponential Distribution : a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital, notation:X~Exp(m). The mean isμ=1m1mand the standard deviation isÏ=1m1m. The probability densi... | https://openstax.org/books/introductory-statistics-2e/pages/7-key-terms |
Mean : a number that measures the central tendency; a common name for mean is "average." The term "mean" is a shortened form of "arithmetic mean." By definition, the mean for a sample (denoted byx¯x¯) isx¯=Sum of all values in the sampleNumber of values in the samplex¯=Sum of all values in the sampleNumber of value... | https://openstax.org/books/introductory-statistics-2e/pages/7-key-terms |
Normal Distribution : a continuous random variable (RV) with pdff(x)=1Ï2Ïeâ(xâμ)22Ï2f(x)=1Ï2Ïeâ(xâμ)22Ï2, whereμis the mean of the distribution andÏis the standard deviation; notation:Χ~N(μ,Ï). Ifμ= 0 andÏ= 1, the RV is called astandard normal distribution. | https://openstax.org/books/introductory-statistics-2e/pages/7-key-terms |
Normal Distribution : a continuous random variable (RV) with pdff(x)=1Ï2Ïeâ(xâμ)22Ï2f(x)=1Ï2Ïeâ(xâμ)22Ï2, whereμis the mean of the distribution andÏis the standard deviation.; notation:X~N(μ,Ï). Ifμ= 0 andÏ= 1, the RV is called thestandard normal distribution. | https://openstax.org/books/introductory-statistics-2e/pages/7-key-terms |
Sampling Distribution : Given simple random samples of sizenfrom a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution. | https://openstax.org/books/introductory-statistics-2e/pages/7-key-terms |
Standard Error of the Mean : the standard deviation of the distribution of the sample means, orÏnÏn. | https://openstax.org/books/introductory-statistics-2e/pages/7-key-terms |
Uniform Distribution : a continuous random variable (RV) that has equally likely outcomes over the domain,a<x<b; often referred as theRectangular Distributionbecause the graph of the pdf has the form of a rectangle. Notation:X~U(a,b). The mean isμ=a+b2μ=a+b2and the standard deviation isÏ=(bâa)212Ï=(bâa)212. The... | https://openstax.org/books/introductory-statistics-2e/pages/7-key-terms |
In this module, we learned how to calculate the confidence interval for a single population mean where the population standard deviation is known. When estimating a population mean, the margin of error is called the error bound for a population mean (EBM). A confidence interval has the general form: | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
(lower bound, upper bound) = (point estimate âEBM, point estimate +EBM) | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
The calculation ofEBMdepends on the size of the sample and the level of confidence desired. The confidence level is the percent of all possible samples that can be expected to include the true population parameter. As the confidence level increases, the correspondingEBMincreases as well. As the sample size increases, t... | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
EBM=zÏnEBM=zÏn | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
Given a confidence interval, you can work backwards to find the error bound (EBM) or the sample mean. To find the error bound, find the difference of the upper bound of the interval and the mean. If you do not know the sample mean, you can find the error bound by calculating half the difference of the upper and lower b... | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
Sometimes researchers know in advance that they want to estimate a population mean within a specific margin of error for a given level of confidence. In that case, solve theEBMformula fornto discover the size of the sample that is needed to achieve this goal: | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
n=z2Ï2EBM2n=z2Ï2EBM2 | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
In many cases, the researcher does not know the population standard deviation,Ï, of the measure being studied. In these cases, it is common to use the sample standard deviation,s, as an estimate ofÏ. The normal distribution creates accurate confidence intervals whenÏis known, but it is not as accurate whensis used a... | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
t=x¯âμsnt=x¯âμsn | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
Thet-score follows the Studentâs t-distribution withnâ 1 degrees of freedom. The confidence interval under this distribution is calculated withEBM=(tα2)sn(tα2)snwheretα2tα2is thet-score with area to the right equal toα2α2,sis the sample standard deviation, andnis the sample size. Use a table, calculator, or c... | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
Some statistical measures, like many survey questions, measure qualitative rather than quantitative data. In this case, the population parameter being estimated is a proportion. It is possible to create a confidence interval for the true population proportion following procedures similar to those used in creating confi... | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
Letpâ²pâ²represent the sample proportion,x/n, wherexrepresents the number of successes andnrepresents the sample size. Letqâ²qâ²= 1 âpâ²pâ². Then the confidence interval for a population proportion is given by the following formula: | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
(lower bound, upper bound)=(pâ²âEBP,pâ²+EBP)=(pâ²âzpâ²qâ²n,pâ²+zpâ²qâ²n)=(pâ²âEBP,pâ²+EBP)=(pâ²âzpâ²qâ²n,pâ²+zpâ²qâ²n) | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
The âplus fourâ method for calculating confidence intervals is an attempt to balance the error introduced by using estimates of the population proportion when calculating the standard deviation of the sampling distribution. Simply imagine four additional trials in the study; two are successes and two are failures. ... | https://openstax.org/books/introductory-statistics-2e/pages/8-chapter-review |
X¯~N(μX,Ïn)X¯~N(μX,Ïn)The distribution of sample means is normally distributed with mean equal to the population mean and standard deviation given by the population standard deviation divided by the square root of the sample size. | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
The general form for a confidence interval for a single population mean, known standard deviation, normal distribution is given by(lower bound, upper bound) = (point estimate âEBM, point estimate +EBM)=(x¯âEBM,x¯+EBM)(x¯âEBM,x¯+EBM)=(x¯âzÏn,x¯+zÏn)(x¯âzÏn,x¯+zÏn) | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
EBM=zÏnzÏn= the error bound for the mean, or the margin of error for a single population mean; this formula is used when the population standard deviation is known. | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
CL= confidence level, or the proportion of confidence intervals created that are expected to contain the true population parameter | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
α= 1 âCL= the proportion of confidence intervals that will not contain the population parameter | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
zα2zα2= thez-score with the property that the area to the right of the z-score isâ2â2this is thez-score used in the calculation of"EBMwhere α = 1 âCL. | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
n=z2Ï2EBM2z2Ï2EBM2= the formula used to determine the sample size (n) needed to achieve a desired margin of error at a given level of confidence | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
General form of a confidence interval | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
(lower value, upper value) = (point estimateâerror bound, point estimate + error bound) | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
To find the error bound when you know the confidence interval | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
error bound = upper valueâpoint estimate OR error bound =upper valueâlower value2upper valueâlower value2 | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
Single Population Mean, Known Standard Deviation, Normal Distribution | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
Use the Normal Distribution for Means, Population Standard Deviation is KnownEBM=zα2âÏnα2âÏn | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
The confidence interval has the format (x¯x¯âEBM,x¯x¯+EBM). | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
s= the standard deviation of sample values. | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
t=x¯âμsnt=x¯âμsnis the formula for thet-score which measures how far away a measure is from the population mean in the Studentâs t-distribution | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
df=n- 1; the degrees of freedom for a Studentâs t-distribution where n represents the size of the sample | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
T~tdfthe random variable,T, has a Studentâs t-distribution withdfdegrees of freedom | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
EBM=tα2snEBM=tα2sn= the error bound for the population mean when the population standard deviation is unknown | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
tα2tα2is thet-score in the Studentâs t-distribution with area to the right equal toα2α2 | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
The general form for a confidence interval for a single mean, population standard deviation unknown, Student's t is given by (lower bound, upper bound)= (point estimate âEBM, point estimate +EBM)=(x¯âtsn,x¯+tsn)(x¯âtsn,x¯+tsn) | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
pâ² = x / nwherexrepresents the number of successes andnrepresents the sample size. The variablepâ² is the sample proportion and serves as the point estimate for the true population proportion. | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
qâ² = 1 âpâ² | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
pâ²~N(p,pqn)pâ²~N(p,pqn)The variablepâ²pâ²has a binomial distribution that can be approximated with the normal distribution shown here. | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
EBP= the error bound for a proportion =zα2pâ²qâ²nzα2pâ²qâ²n | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
Confidence interval for a proportion: | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
(lower bound, upper bound)=(pâ²âEBP,pâ²+EBP)=(pâ²âzpâ²qâ²n,pâ²+zpâ²qâ²n)=(pâ²âEBP,pâ²+EBP)=(pâ²âzpâ²qâ²n,pâ²+zpâ²qâ²n) | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
n=zα22pâ²qâ²EBP2n=zα22pâ²qâ²EBP2provides the number of participants needed to estimate the population proportion with confidence 1 -αand margin of errorEBP. | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
Use the normal distribution for a single population proportionpâ²=xnpâ²=xn | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
EBP=(zα2)pâ²qâ²npâ²+qâ²=1EBP=(zα2)pâ²qâ²npâ²+qâ²=1 | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
The confidence interval has the format (pâ²pâ²âEBP,pâ²pâ²+EBP). | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
x¯x¯is a point estimate forμ | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
pâ²pâ²is a point estimate forÏ | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
sis a point estimate forÏ | https://openstax.org/books/introductory-statistics-2e/pages/8-formula-review |
Binomial Distribution : a discrete random variable (RV) which arises from Bernoulli trials; there are a fixed number,n, of independent trials. âIndependentâ means that the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same cond... | https://openstax.org/books/introductory-statistics-2e/pages/8-key-terms |
Confidence Interval (CI) : an interval estimate for an unknown population parameter. This depends on:the desired confidence level,information that is known about the distribution (for example, known standard deviation),the sample and its size. | https://openstax.org/books/introductory-statistics-2e/pages/8-key-terms |
Confidence Level (CL) : the percent expression for the probability that the confidence interval contains the true population parameter; for example, if the CL = 90%, then in 90 out of 100 samples the interval estimate will enclose the true population parameter. | https://openstax.org/books/introductory-statistics-2e/pages/8-key-terms |
Degrees of Freedom (df) : the number of objects in a sample that are free to vary | https://openstax.org/books/introductory-statistics-2e/pages/8-key-terms |
Error Bound for a Population Mean (EBM) : the margin of error; depends on the confidence level, sample size, and known or estimated population standard deviation. | https://openstax.org/books/introductory-statistics-2e/pages/8-key-terms |
Inferential Statistics : also called statistical inference or inductive statistics; this facet of statistics deals with estimating a population parameter based on a sample statistic. For example, if four out of the 100 calculators sampled are defective we might infer that four percent of the production is defective. | https://openstax.org/books/introductory-statistics-2e/pages/8-key-terms |
Normal Distribution : a continuous random variable (RV) with pdff(x)=1Ï2Ïeâ(xâμ)2/2Ï2f(x)=1Ï2Ïeâ(xâμ)2/2Ï2, whereμis the mean of the distribution andÏis the standard deviation, notation:X~N(μ,Ï). Ifμ= 0 andÏ= 1, the RV is calledthe standard normal distribution. | https://openstax.org/books/introductory-statistics-2e/pages/8-key-terms |
Parameter : a numerical characteristic of a population | https://openstax.org/books/introductory-statistics-2e/pages/8-key-terms |
Point Estimate : a single number computed from a sample and used to estimate a population parameter | https://openstax.org/books/introductory-statistics-2e/pages/8-key-terms |
Standard Deviation : a number that is equal to the square root of the variance and measures how far data values are from their mean; notation:sfor sample standard deviation andÏfor population standard deviation | https://openstax.org/books/introductory-statistics-2e/pages/8-key-terms |
Student'st-Distribution : investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student; the major characteristics of the random variable (RV) are:It is continuous and assumes any real values.The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at... | https://openstax.org/books/introductory-statistics-2e/pages/8-key-terms |
In ahypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
Evaluate thenull hypothesis, typically denoted withH0. The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ⤠or â¥) | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
Always write thealternative hypothesis, typically denoted withHaorH1, using less than, greater than, or not equals symbols, i.e., (â , >, or <). | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties. | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
In every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or misunderstood summary statistics can yield errors that affect the results. AType Ierror occurs when a true null hypothesis is rejected. AType II erroroccurs when a false null hypothesis is not rejecte... | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
The probabilities of these errors are denoted by the Greek lettersαandβ, for a Type I and a Type II error respectively. The power of the test, 1 âβ, quantifies the likelihood that a test will yield the correct result of a true alternative hypothesis being accepted. A high power is desirable. | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
In order for a hypothesis testâs results to be generalized to a population, certain requirements must be satisfied. | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
When testing for a single population mean: | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
A Student'st-test should be used if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with an unknown standard deviation. | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
The normal test will work if the data come from a simple, random sample and the population
is approximately normally distributed, or the sample size is large, with a known standard
deviation. | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
When testing a single population proportion use a normal test for a single population proportion if the data comes from a simple, random sample, fill the requirements for a binomial distribution, and the mean number of successes and the mean number of failures satisfy the conditions:np> 5 andnq> 5 wherenis the sample s... | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
When the probability of an event occurring is low, and it happens, it is called a rare event. Rare events are important to consider in hypothesis testing because they can inform your willingness not to reject or to reject a null hypothesis. To test a null hypothesis, find thep-value for the sample data and graph the re... | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
α>p-value, reject the null hypothesis | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
αâ¤p-value, do not reject the null hypothesis | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
Thehypothesis testitself has an established process. This can be summarized as follows: | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
DetermineH0andHa. Remember, they are contradictory. | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
Determine the random variable. | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
Determine the distribution for the test. | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
Draw a graph, calculate the test statistic, and use the test statistic to calculate thep-value. (Az-score and at-score are examples of test statistics.) | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
Compare the preconceivedαwith thep-value, make a decision (reject or do not rejectH0), and write a clear conclusion using English sentences. | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
Notice that in performing the hypothesis test, you useαand notβ.βis needed to help determine the sample size of the data that is used in calculating thep-value. Remember that the quantity 1 âβis called thePower of the Test. A high power is desirable. If the power is too low, statisticians typically increase the s... | https://openstax.org/books/introductory-statistics-2e/pages/9-chapter-review |
equal (=) | https://openstax.org/books/introductory-statistics-2e/pages/9-formula-review |
Binomial Distribution : a discrete random variable (RV) that arises from Bernoulli trials. There are a fixed number,n, of independent trials. âIndependentâ means that the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same condi... | https://openstax.org/books/introductory-statistics-2e/pages/9-key-terms |
Central Limit Theorem : Given a random variable (RV) with known meanμμand known standard deviation Ï. We are sampling with sizenand we are interested in two new RVs - the sample mean,X¯X¯, and the sample sum,ΣXΣX.
If the sizenof the sample is sufficiently large, thenX¯~N(μ,Ïn)X¯~N(μ,Ïn)andΣX~N(nμ,nÏ)ΣX... | https://openstax.org/books/introductory-statistics-2e/pages/9-key-terms |
Confidence Interval (CI) : an interval estimate for an unknown population parameter. This depends on:The desired confidence level.Information that is known about the distribution (for example, known standard deviation).The sample and its size. | https://openstax.org/books/introductory-statistics-2e/pages/9-key-terms |
Hypothesis : a statement about the value of a population parameter, in case of two hypotheses, the statement assumed to be true is called the null hypothesis (notationH0) and the contradictory statement is called the alternative hypothesis (notationHa). | https://openstax.org/books/introductory-statistics-2e/pages/9-key-terms |
Hypothesis Testing : Based on sample evidence, a procedure for determining whether the hypothesis stated is a reasonable statement and should not be rejected, or is unreasonable and should be rejected. | https://openstax.org/books/introductory-statistics-2e/pages/9-key-terms |
Level of Significance of the Test : probability of a Type I error (reject the null hypothesis when it is true). Notation: α. In hypothesis testing, the Level of Significance is called the preconceived α or the preset α. | https://openstax.org/books/introductory-statistics-2e/pages/9-key-terms |
Normal Distribution : a continuous random variable (RV) with pdff(x)=1Ï2Ïeâ(xâμ)22Ï2f(x)=1Ï2Ïeâ(xâμ)22Ï2, whereμis the mean of the distribution, andÏis the standard deviation, notation:X ~ N(μ,Ï). Ifμ= 0 andÏ= 1, the RV is calledthe standard normal distribution. | https://openstax.org/books/introductory-statistics-2e/pages/9-key-terms |
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