Data Sets Lap Times The following tables provide lap times from Terri Vogel's log book. Times are recorded in seconds for 2.5-mile laps completed in a series of races and practice runs. Race Lap Times (in seconds) Lap 1 Lap 2 Lap 3 Lap 4 Lap 5 Lap 6 Lap 7 Race 1 135 130 131 132 130 131 133 Race 2 134 131 131 129 128 128 129 Race 3 129 128 127 127 130 127 129 Race 4 125 125 126 125 124 125 125 Race 5 133 132 132 132 131 130 132 Race 6 130 130 130 129 129 130 129 Race 7 132 131 133 131 134 134 131 Race 8 127 128 127 130 128 126 128 Race 9 132 130 127 128 126 127 124 Race 10 135 131 131 132 130 131 130 Race 11 132 131 132 131 130 129 129 Race 12 134 130 130 130 131 130 130 Race 13 128 127 128 128 128 129 128 Race 14 132 131 131 131 132 130 130 Race 15 136 129 129 129 129 129 129 Race 16 129 129 129 128 128 129 129 Race 17 134 131 132 131 132 132 132 Race 18 129 129 130 130 133 133 127 Race 19 130 129 129 129 129 129 128 Race 20 131 128 130 128 129 130 130 Practice Lap Times (in seconds) Lap 1 Lap 2 Lap 3 Lap 4 Lap 5 Lap 6 Lap 7 Practice 1 142 143 180 137 134 134 172 Practice 2 140 135 134 133 128 128 131 Practice 3 130 133 130 128 135 133 133 Practice 4 141 136 137 136 136 136 145 Practice 5 140 138 136 137 135 134 134 Practice 6 142 142 139 138 129 129 127 Practice 7 139 137 135 135 137 134 135 Practice 8 143 136 134 133 134 133 132 Practice 9 135 134 133 133 132 132 133 Practice 10 131 130 128 129 127 128 127 Practice 11 143 139 139 138 138 137 138 Practice 12 132 133 131 129 128 127 126 Practice 13 149 144 144 139 138 138 137 Practice 14 133 132 137 133 134 130 131 Practice 15 138 136 133 133 132 131 131 Stock Prices The following table lists initial public offering (IPO) stock prices for all 1999 stocks that at least doubled in value during the first day of trading. IPO Offer Prices $17.00 $23.00 $14.00 $16.00 $12.00 $26.00 $20.00 $22.00 $14.00 $15.00 $22.00 $18.00 $18.00 $21.00 $21.00 $19.00 $15.00 $21.00 $18.00 $17.00 $15.00 $25.00 $14.00 $30.00 $16.00 $10.00 $20.00 $12.00 $16.00 $17.44 $16.00 $14.00 $15.00 $20.00 $20.00 $16.00 $17.00 $16.00 $15.00 $15.00 $19.00 $48.00 $16.00 $18.00 $9.00 $18.00 $18.00 $20.00 $8.00 $20.00 $17.00 $14.00 $11.00 $16.00 $19.00 $15.00 $21.00 $12.00 $8.00 $16.00 $13.00 $14.00 $15.00 $14.00 $13.41 $28.00 $21.00 $17.00 $28.00 $17.00 $19.00 $16.00 $17.00 $19.00 $18.00 $17.00 $15.00 $14.00 $21.00 $12.00 $18.00 $24.00 $15.00 $23.00 $14.00 $16.00 $12.00 $24.00 $20.00 $14.00 $14.00 $15.00 $14.00 $19.00 $16.00 $38.00 $20.00 $24.00 $16.00 $8.00 $18.00 $17.00 $16.00 $15.00 $7.00 $19.00 $12.00 $8.00 $23.00 $12.00 $18.00 $20.00 $21.00 $34.00 $16.00 $26.00 $14.00 References Data compiled by Jay R. Ritter of University of Florida using data from Securities Data Co. and Bloomberg .

Math


Group and Partner Projects Univariate Data Student Learning Objectives The student will design and carry out a survey. The student will analyze and graphically display the results of the survey. Instructions As you complete each task below, check it off. Answer all questions in your summary. ____ Decide what data you are going to study. Here are two examples, but you may NOT use them: number of M&M's per bag, number of pencils students have in their backpacks. ____ Are your data discrete or continuous? How do you know? ____ Decide how you are going to collect the data (for instance, buy 30 bags of M&M's; collect data from the World Wide Web). ____ Describe your sampling technique in detail. Use cluster, stratified, systematic, or simple random (using a random number generator) sampling. Do not use convenience sampling. Which method did you use? Why did you pick that method? ____ Conduct your survey. Your data size must be at least 30. ____ Summarize your data in a chart with columns showing data value, frequency, relative frequency and cumulative relative frequency. Answer the following (rounded to two decimal places): x ¯ = _____ s = _____ First quartile = _____ Median = _____ 70 th percentile = _____ ____ What value is two standard deviations above the mean? ____ What value is 1.5 standard deviations below the mean? ____ Construct a histogram displaying your data. ____ In complete sentences, describe the shape of your graph. ____ Do you notice any potential outliers? If so, what values are they? Show your work in how you used the potential outlier formula to determine whether or not the values might be outliers. ____ Construct a box plot displaying your data. ____ Does the middle 50% of the data appear to be concentrated together or spread apart? Explain how you determined this. ____ Looking at both the histogram and the box plot, discuss the distribution of your data. Assignment Checklist You need to turn in the following typed and stapled packet, with pages in the following order: ____ Cover sheet : name, class time, and name of your study ____ Summary page : This should contain paragraphs written with complete sentences. It should include answers to all the questions above. It should also include statements describing the population under study, the sample, a parameter or parameters being studied, and the statistic or statistics produced. ____ URL for data, if your data are from the World Wide Web ____ Chart of data, frequency, relative frequency, and cumulative relative frequency ____ Page(s) of graphs: histogram and box plot Continuous Distributions and Central Limit Theorem Student Learning Objectives The student will collect a sample of continuous data. The student will attempt to fit the data sample to various distribution models. The student will validate the central limit theorem. Instructions As you complete each task below, check it off. Answer all questions in your summary. Part I: Sampling ____ Decide what continuous data you are going to study. (Here are two examples, but you may NOT use them: the amount of money a student spent on college supplies this term, or the length of time distance telephone call lasts.) ____ Describe your sampling technique in detail. Use cluster, stratified, systematic, or simple random (using a random number generator) sampling. Do not use convenience sampling. What method did you use? Why did you pick that method? ____ Conduct your survey. Gather at least 150 pieces of continuous, quantitative data . ____ Define (in words) the random variable for your data. X = _______ ____ Create two lists of your data: (1) unordered data, (2) in order of smallest to largest. ____ Find the sample mean and the sample standard deviation (rounded to two decimal places). x ¯ = ______ s = ______ ____ Construct a histogram of your data containing five to ten intervals of equal width. The histogram should be a representative display of your data. Label and scale it. Part II: Possible Distributions ____ Suppose that X followed the following theoretical distributions. Set up each distribution using the appropriate information from your data. ____ Uniform: X ~ U ____________ Use the lowest and highest values as a and b . ____ Normal: X ~ N ____________ Use x ¯ to estimate for μ and s to estimate for σ . ____ Must your data fit one of the above distributions? Explain why or why not. ____ Could the data fit two or three of the previous distributions (at the same time)? Explain. ____ Calculate the value k (an X value) that is 1.75 standard deviations above the sample mean. k = _________ (rounded to two decimal places) Note: k = x ¯ + (1.75) s ____ Determine the relative frequencies ( RF ) rounded to four decimal places. NOTE R F = frequency total number surveyed RF ( X < k ) = ______ RF ( X > k ) = ______ RF ( X = k ) = ______ NOTE You should have one page for the uniform distribution, one page for the exponential distribution, and one page for the normal distribution. ____ State the distribution: X ~ _________ ____ Draw a graph for each of the three theoretical distributions. Label the axes and mark them appropriately. ____ Find the following theoretical probabilities (rounded to four decimal places). P ( X < k ) = ______ P ( X > k ) = ______ P ( X = k ) = ______ ____ Compare the relative frequencies to the corresponding probabilities. Are the values close? ____ Does it appear that the data fit the distribution well? Justify your answer by comparing the probabilities to the relative frequencies, and the histograms to the theoretical graphs. Part III: CLT Experiments ______ From your original data (before ordering), use a random number generator to pick 40 samples of size five. For each sample, calculate the average. ______ On a separate page, attached to the summary, include the 40 samples of size five, along with the 40 sample averages. ______ List the 40 averages in order from smallest to largest. ______ Define the random variable, X ¯ , in words. X ¯ = _______________ ______ State the approximate theoretical distribution of X ¯ . X ¯ ~ ______________ ______ Base this on the mean and standard deviation from your original data. ______ Construct a histogram displaying your data. Use five to six intervals of equal width. Label and scale it. Calculate the value k ¯ (an X ¯ value) that is 1.75 standard deviations above the sample mean. k ¯ = _____ (rounded to two decimal places) Determine the relative frequencies ( RF ) rounded to four decimal places. RF ( X ¯ < k ¯ ) = _______ RF ( X ¯ > k ¯ ) = _______ RF ( X ¯ = k ¯ ) = _______ Find the following theoretical probabilities (rounded to four decimal places). P ( X ¯ < k ¯ ) = _______ P ( X ¯ > k ¯ ) = _______ P ( X ¯ = k ¯ ) = _______ ______ Draw the graph of the theoretical distribution of X . ______ Compare the relative frequencies to the probabilities. Are the values close? ______ Does it appear that the data of averages fit the distribution of X ¯ well? Justify your answer by comparing the probabilities to the relative frequencies, and the histogram to the theoretical graph. In three to five complete sentences for each, answer the following questions. Give thoughtful explanations. ______ In summary, do your original data seem to fit the uniform, exponential, or normal distributions? Answer why or why not for each distribution. If the data do not fit any of those distributions, explain why. ______ What happened to the shape and distribution when you averaged your data? In theory, what should have happened? In theory, would “it” always happen? Why or why not? ______ Were the relative frequencies compared to the theoretical probabilities closer when comparing the X or X ¯ distributions? Explain your answer. Assignment Checklist You need to turn in the following typed and stapled packet, with pages in the following order: ____ Cover sheet : name, class time, and name of your study ____ Summary pages : These should contain several paragraphs written with complete sentences that describe the experiment, including what you studied and your sampling technique, as well as answers to all of the questions previously asked questions ____ URL for data, if your data are from the World Wide Web ____ Pages, one for each theoretical distribution , with the distribution stated, the graph, and the probability questions answered ____ Pages of the data requested ____ All graphs required Hypothesis Testing-Article Student Learning Objectives The student will identify a hypothesis testing problem in print. The student will conduct a survey to verify or dispute the results of the hypothesis test. The student will summarize the article, analysis, and conclusions in a report. Instructions As you complete each task, check it off. Answer all questions in your summary. ____ Find an article in a newspaper, magazine, or on the internet which makes a claim about ONE population mean or ONE population proportion. The claim may be based upon a survey that the article was reporting on. Decide whether this claim is the null or alternate hypothesis. ____ Copy or print out the article and include a copy in your project, along with the source. ____ State how you will collect your data. (Convenience sampling is not acceptable.) ____ Conduct your survey. You must have more than 50 responses in your sample. When you hand in your final project, attach the tally sheet or the packet of questionnaires that you used to collect data. Your data must be real. ____ State the statistics that are a result of your data collection: sample size, sample mean, and sample standard deviation, OR sample size and number of successes. ____ Make two copies of the appropriate solution sheet. ____ Record the hypothesis test on the solution sheet, based on your experiment. Do a DRAFT solution first on one of the solution sheets and check it over carefully. Have a classmate check your solution to see if it is done correctly. Make your decision using a 5% level of significance. Include the 95% confidence interval on the solution sheet. ____ Create a graph that illustrates your data. This may be a pie or bar graph or may be a histogram or box plot, depending on the nature of your data. Produce a graph that makes sense for your data and gives useful visual information about your data. You may need to look at several types of graphs before you decide which is the most appropriate for the type of data in your project. ____ Write your summary (in complete sentences and paragraphs, with proper grammar and correct spelling) that describes the project. The summary MUST include: Brief discussion of the article, including the source Statement of the claim made in the article (one of the hypotheses). Detailed description of how, where, and when you collected the data, including the sampling technique; did you use cluster, stratified, systematic, or simple random sampling (using a random number generator)? As previously mentioned, convenience sampling is not acceptable. Conclusion about the article claim in light of your hypothesis test; this is the conclusion of your hypothesis test, stated in words, in the context of the situation in your project in sentence form, as if you were writing this conclusion for a non-statistician. Sentence interpreting your confidence interval in the context of the situation in your project Assignment Checklist Turn in the following typed (12 point) and stapled packet for your final project: ____ Cover sheet containing your name(s), class time, and the name of your study ____ Summary , which includes all items listed on summary checklist ____ Solution sheet neatly and completely filled out. The solution sheet does not need to be typed. ____ Graphic representation of your data , created following the guidelines previously discussed; include only graphs which are appropriate and useful. ____ Raw data collected AND a table summarizing the sample data ( n , x ¯ and s ; or x , n , and p ’, as appropriate for your hypotheses); the raw data does not need to be typed, but the summary does. Hand in the data as you collected it. (Either attach your tally sheet or an envelope containing your questionnaires.) Bivariate Data, Linear Regression, and Univariate Data Student Learning Objectives The students will collect a bivariate data sample through the use of appropriate sampling techniques. The student will attempt to fit the data to a linear model. The student will determine the appropriateness of linear fit of the model. The student will analyze and graph univariate data. Instructions As you complete each task below, check it off. Answer all questions in your introduction or summary. Check your course calendar for intermediate and final due dates. Graphs may be constructed by hand or by computer, unless your instructor informs you otherwise. All graphs must be neat and accurate. All other responses must be done on the computer. Neatness and quality of explanations are used to determine your final grade. Part I: Bivariate Data Introduction ____State the bivariate data your group is going to study. Here are two examples, but you may NOT use them: height vs. weight and age vs. running distance. ____Describe your sampling technique in detail. Use cluster, stratified, systematic, or simple random sampling (using a random number generator) sampling. Convenience sampling is NOT acceptable. ____Conduct your survey. Your number of pairs must be at least 30. ____Print out a copy of your data. Analysis ____On a separate sheet of paper construct a scatter plot of the data. Label and scale both axes. ____State the least squares line and the correlation coefficient. ____On your scatter plot, in a different color, construct the least squares line. ____Is the correlation coefficient significant? Explain and show how you determined this. ____Interpret the slope of the linear regression line in the context of the data in your project. Relate the explanation to your data, and quantify what the slope tells you. ____Does the regression line seem to fit the data? Why or why not? If the data does not seem to be linear, explain if any other model seems to fit the data better. ____Are there any outliers? If so, what are they? Show your work in how you used the potential outlier formula in the Linear Regression and Correlation chapter (since you have bivariate data) to determine whether or not any pairs might be outliers. Part II: Univariate Data In this section, you will use the data for ONE variable only. Pick the variable that is more interesting to analyze. For example: if your independent variable is sequential data such as year with 30 years and one piece of data per year, your x -values might be 1971, 1972, 1973, 1974, …, 2000. This would not be interesting to analyze. In that case, choose to use the dependent variable to analyze for this part of the project. _____Summarize your data in a chart with columns showing data value, frequency, relative frequency, and cumulative relative frequency. _____Answer the following question, rounded to two decimal places: Sample mean = ______ Sample standard deviation = ______ First quartile = ______ Third quartile = ______ Median = ______ 70th percentile = ______ Value that is 2 standard deviations above the mean = ______ Value that is 1.5 standard deviations below the mean = ______ _____Construct a histogram displaying your data. Group your data into six to ten intervals of equal width. Pick regularly spaced intervals that make sense in relation to your data. For example, do NOT group data by age as 20-26,27-33,34-40,41-47,48-54,55-61 . . . Instead, maybe use age groups 19.5-24.5, 24.5-29.5, . . . or 19.5-29.5, 29.5-39.5, 39.5-49.5, . . . _____In complete sentences, describe the shape of your histogram. _____Are there any potential outliers? Which values are they? Show your work and calculations as to how you used the potential outlier formula in Descriptive Statistics (since you are now using univariate data) to determine which values might be outliers. _____Construct a box plot of your data. _____Does the middle 50% of your data appear to be concentrated together or spread out? Explain how you determined this. _____Looking at both the histogram AND the box plot, discuss the distribution of your data. For example: how does the spread of the middle 50% of your data compare to the spread of the rest of the data represented in the box plot; how does this correspond to your description of the shape of the histogram; how does the graphical display show any outliers you may have found; does the histogram show any gaps in the data that are not visible in the box plot; are there any interesting features of your data that you should point out. Due Dates Part I, Intro: __________ (keep a copy for your records) Part I, Analysis: __________ (keep a copy for your records) Entire Project, typed and stapled: __________ ____ Cover sheet: names, class time, and name of your study ____ Part I: label the sections “Intro” and “Analysis.” ____ Part II: ____ Summary page containing several paragraphs written in complete sentences describing the experiment, including what you studied and how you collected your data. The summary page should also include answers to ALL the questions asked above. ____ All graphs requested in the project ____ All calculations requested to support questions in data ____ Description: what you learned by doing this project, what challenges you had, how you overcame the challenges NOTE Include answers to ALL questions asked, even if not explicitly repeated in the items above.

Math


Solution Sheets Hypothesis Testing with One Sample Class Time: __________________________ Name: _____________________________________ H 0 : _______ H a : _______ In words, CLEARLY state what your random variable X ¯ or P ′ represents. State the distribution to use for the test. What is the test statistic? What is the p -value? In one or two complete sentences, explain what the p -value means for this problem. Use the previous information to sketch a picture of this situation. CLEARLY, label and scale the horizontal axis and shade the region(s) corresponding to the p -value. Indicate the correct decision (“reject” or “do not reject” the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences . Alpha: _______ Decision: _______ Reason for decision: _______ Conclusion: _______ Construct a 95% confidence interval for the true mean or proportion. Include a sketch of the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval. Hypothesis Testing with Two Samples Class Time: __________________________ Name: _____________________________________ H 0 : _______ H a : _______ In words, clearly state what your random variable X ¯ 1 − X ¯ 2 , P ′ 1 − P ′ 2 or X ¯ d represents. State the distribution to use for the test. What is the test statistic? What is the p -value? In one to two complete sentences, explain what the p-value means for this problem. Use the previous information to sketch a picture of this situation. CLEARLY label and scale the horizontal axis and shade the region(s) corresponding to the p -value. Indicate the correct decision (“reject” or “do not reject” the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences . Alpha: _______ Decision: _______ Reason for decision: _______ Conclusion: _______ In complete sentences, explain how you determined which distribution to use. The Chi-Square Distribution Class Time: __________________________ Name: ____________________________________ H 0 : _______ H a : _______ What are the degrees of freedom? State the distribution to use for the test. What is the test statistic? What is the p -value? In one to two complete sentences, explain what the p -value means for this problem. Use the previous information to sketch a picture of this situation. Clearly label and scale the horizontal axis and shade the region(s) corresponding to the p -value. Indicate the correct decision (“reject” or “do not reject” the null hypothesis) and write appropriate conclusions, using complete sentences. Alpha: _______ Decision: _______ Reason for decision: _______ Conclusion: _______ F Distribution and One-Way ANOVA Class Time: __________________________ Name: ____________________________________ H 0 : _______ H a : _______ df ( n ) = ______ df ( d ) = _______ State the distribution to use for the test. What is the test statistic? What is the p -value? Use the previous information to sketch a picture of this situation. Clearly label and scale the horizontal axis and shade the region(s) corresponding to the p -value. Indicate the correct decision (“reject” or “do not reject” the null hypothesis) and write appropriate conclusions, using complete sentences . Alpha: _______ Decision: _______ Reason for decision: _______ Conclusion: _______

Math


Mathematical Phrases, Symbols, and Formulas English Phrases Written Mathematically When the English says: Interpret this as: X is at least 4. X ≥ 4 The minimum of X is 4. X ≥ 4 X is no less than 4. X ≥ 4 X is greater than or equal to 4. X ≥ 4 X is at most 4. X ≤ 4 The maximum of X is 4. X ≤ 4 X is no more than 4. X ≤ 4 X is less than or equal to 4. X ≤ 4 X does not exceed 4. X ≤ 4 X is greater than 4. X > 4 X is more than 4. X > 4 X exceeds 4. X > 4 X is less than 4. X < 4 There are fewer X than 4. X < 4 X is 4. X = 4 X is equal to 4. X = 4 X is the same as 4. X = 4 X is not 4. X ≠ 4 X is not equal to 4. X ≠ 4 X is not the same as 4. X ≠ 4 X is different than 4. X ≠ 4 Formulas Formula 1: Factorial n ! = n ( n − 1 ) ( n − 2 ) . . . ( 1 ) 0 ! = 1 Formula 2: Combinations ( n r ) = n ! ( n − r ) ! r ! Formula 3: Binomial Distribution X ~ B ( n , p ) P ( X = x ) = ( n x ) p x q n − x , for x = 0 , 1 , 2 , . . . , n Formula 4: Geometric Distribution X ~ G ( p ) P ( X = x ) = q x − 1 p , for x = 1 , 2 , 3 , . . . Formula 5: Hypergeometric Distribution X ~ H ( r , b , n ) P ( X = x ) = ( ( r x ) ( b n − x ) ( r + b n ) ) Formula 6: Poisson Distribution X ~ P ( μ ) P ( X = x ) = μ x e − μ x ! Formula 7: Uniform Distribution X ~ U ( a , b ) f ( X ) = 1 b − a , a < x < b Formula 8: Exponential Distribution X ~ E x p ( m ) f ( x ) = m e − m x m > 0 , x ≥ 0 Formula 9: Normal Distribution X ~ N ( μ , σ 2 ) f ( x ) = 1 σ 2 π e − ( x − μ ) 2 2 σ 2 , – ∞ < x < ∞ Formula 10: Student's t -distribution X ~ t d f f ( x ) = ( 1 + x 2 n ) − ( n + 1 ) 2 Γ ( n + 1 2 ) nπ Γ ( n 2 ) X = Z Y n Z ~ N ( 0 , 1 ), Y ~ Χ d f 2 , n = degrees of freedom Formula 11: Chi-Square Distribution X ~ Χ d f 2 f ( x ) = x n − 2 2 e − x 2 2 n 2 Γ ( n 2 ) , x > 0 , n = positive integer and degrees of freedom Formula 12: F Distribution X ~ F d f ( n ) , d f ( d ) d f ( n ) = degrees of freedom for the numerator d f ( d ) = degrees of freedom for the denominator f ( x ) = Γ ( u + v 2 ) Γ ( u 2 ) Γ ( v 2 ) ( u v ) u 2 x ( u 2 − 1 ) [ 1 + ( u v ) x − 0.5 ( u + v ) ] X = Y u W v , Y , W are chi-square Symbols and Their Meanings Symbols and their Meanings Chapter (1st used) Symbol Spoken Meaning Sampling and Data The square root of same Sampling and Data π Pi 3.14159… (a specific number) Descriptive Statistics Q 1 Quartile one the first quartile Descriptive Statistics Q 2 Quartile two the second quartile Descriptive Statistics Q 3 Quartile three the third quartile Descriptive Statistics IQR interquartile range Q 3 – Q 1 = IQR Descriptive Statistics x ¯ x-bar sample mean Descriptive Statistics μ mu population mean Descriptive Statistics s s x sx s sample standard deviation Descriptive Statistics s 2 s x 2 s squared sample variance Descriptive Statistics σ σ x σx sigma population standard deviation Descriptive Statistics σ 2 σ x 2 sigma squared population variance Descriptive Statistics Σ capital sigma sum Probability Topics { } brackets set notation Probability Topics S S sample space Probability Topics A Event A event A Probability Topics P ( A ) probability of A probability of A occurring Probability Topics P ( A | B ) probability of A given B prob. of A occurring given B has occurred Probability Topics P ( A OR B ) prob. of A or B prob. of A or B or both occurring Probability Topics P ( A AND B ) prob. of A and B prob. of both A and B occurring (same time) Probability Topics A ′ A-prime, complement of A complement of A, not A Probability Topics P ( A ') prob. of complement of A same Probability Topics G 1 green on first pick same Probability Topics P ( G 1 ) prob. of green on first pick same Discrete Random Variables PDF prob. distribution function same Discrete Random Variables X X the random variable X Discrete Random Variables X ~ the distribution of X same Discrete Random Variables B binomial distribution same Discrete Random Variables G geometric distribution same Discrete Random Variables H hypergeometric dist. same Discrete Random Variables P Poisson dist. same Discrete Random Variables λ Lambda average of Poisson distribution Discrete Random Variables ≥ greater than or equal to same Discrete Random Variables ≤ less than or equal to same Discrete Random Variables = equal to same Discrete Random Variables ≠ not equal to same Continuous Random Variables f ( x ) f of x function of x Continuous Random Variables pdf prob. density function same Continuous Random Variables U uniform distribution same Continuous Random Variables Exp exponential distribution same Continuous Random Variables k k critical value Continuous Random Variables f ( x ) = f of x equals same Continuous Random Variables m m decay rate (for exp. dist.) The Normal Distribution N normal distribution same The Normal Distribution z z -score same The Normal Distribution Z standard normal dist. same The Central Limit Theorem CLT Central Limit Theorem same The Central Limit Theorem X ¯ X -bar the random variable X -bar The Central Limit Theorem μ x mean of X the average of X The Central Limit Theorem μ x ¯ mean of X -bar the average of X -bar The Central Limit Theorem σ x standard deviation of X same The Central Limit Theorem σ x ¯ standard deviation of X -bar same The Central Limit Theorem Σ X sum of X same The Central Limit Theorem Σ x sum of x same Confidence Intervals CL confidence level same Confidence Intervals CI confidence interval same Confidence Intervals EBM error bound for a mean same Confidence Intervals EBP error bound for a proportion same Confidence Intervals t Student's t -distribution same Confidence Intervals df degrees of freedom same Confidence Intervals t α 2 student t with a /2 area in right tail same Confidence Intervals p ′ ; p ^ p -prime; p -hat sample proportion of success Confidence Intervals q ′ ; q ^ q -prime; q -hat sample proportion of failure Hypothesis Testing H 0 H -naught, H -sub 0 null hypothesis Hypothesis Testing H a H-a , H -sub a alternate hypothesis Hypothesis Testing H 1 H -1, H -sub 1 alternate hypothesis Hypothesis Testing α alpha probability of Type I error Hypothesis Testing β beta probability of Type II error Hypothesis Testing X 1 ¯ − X 2 ¯ X 1-bar minus X 2-bar difference in sample means Hypothesis Testing μ 1 − μ 2 mu -1 minus mu -2 difference in population means Hypothesis Testing P ′ 1 − P ′ 2 P 1-prime minus P 2-prime difference in sample proportions Hypothesis Testing p 1 − p 2 p 1 minus p 2 difference in population proportions Chi-Square Distribution Χ 2 Ky -square Chi-square Chi-Square Distribution O Observed Observed frequency Chi-Square Distribution E Expected Expected frequency Linear Regression and Correlation y = a + bx y equals a plus b-x equation of a line Linear Regression and Correlation y ^ y -hat estimated value of y Linear Regression and Correlation r "r" sample correlation coefficient Linear Regression and Correlation ρ rho ("row") population correlation coefficient Linear Regression and Correlation ε error same Linear Regression and Correlation SSE Sum of Squared Errors same Linear Regression and Correlation 1.9 s 1.9 times s cut-off value for outliers F -Distribution and ANOVA F F -ratio F -ratio

Math


NOTEs for the TI-83, 83+, 84, 84+ Calculators Quick Tips Legend represents a button press [ ] represents yellow command or green letter behind a key < > represents items on the screen To adjust the contrast Press , then hold to increase the contrast or to decrease the contrast. To capitalize letters and words Press to get one capital letter, or press , then to set all button presses to capital letters. You can return to the top-level button values by pressing again. To correct a mistake If you hit a wrong button, just hit and start again. To write in scientific notation Numbers in scientific notation are expressed on the TI-83, 83+, 84, and 84+ using E notation, such that... 4.321 E 4 = 4 .321 × 10 4 4.321 E –4 = 4 .321 × 10 –4 To transfer programs or equations from one calculator to another: Both calculators: Insert your respective end of the link cable cable and press , then [LINK] . Calculator receiving information: Use the arrows to navigate to and select <RECEIVE> Press . Calculator sending information: Press appropriate number or letter. Use up and down arrows to access the appropriate item. Press to select item to transfer. Press right arrow to navigate to and select <TRANSMIT> . Press . NOTE ERROR 35 LINK generally means that the cables have not been inserted far enough. Both calculators: Insert your respective end of the link cable cable Both calculators: press , then [QUIT] to exit when done. Manipulating One-Variable Statistics NOTE These directions are for entering data with the built-in statistical program. Sample Data We are manipulating one-variable statistics. Data Frequency –2 10 –1 3 0 4 1 5 3 8 To begin: Turn on the calculator. Access statistics mode. Select <4:ClrList> to clear data from lists, if desired. , Enter list [L1] to be cleared. , [L1] , Display last instruction. , [ENTRY] Continue clearing remaining lists in the same fashion, if desired. , , [L2] , Access statistics mode. Select <1:Edit . . .> Enter data. Data values go into [L1] . (You may need to arrow over to [L1] ). Type in a data value and enter it. (For negative numbers, use the negate (-) key at the bottom of the keypad). , , Continue in the same manner until all data values are entered. In [L2] , enter the frequencies for each data value in [L1] . Type in a frequency and enter it. (If a data value appears only once, the frequency is "1"). , Continue in the same manner until all data values are entered. Access statistics mode. Navigate to <CALC> . Access <1:1-var Stats> . Indicate that the data is in [L1] ... , [L1] , ...and indicate that the frequencies are in [L2] . , [L2] , The statistics should be displayed. You may arrow down to get remaining statistics. Repeat as necessary. Drawing Histograms NOTE We will assume that the data is already entered. We will construct two histograms with the built-in STATPLOT application. The first way will use the default ZOOM. The second way will involve customizing a new graph. Access graphing mode. , [STAT PLOT] Select <1:plot 1> to access plotting - first graph. Use the arrows navigate go to <ON> to turn on Plot 1. <ON> , Use the arrows to go to the histogram picture and select the histogram. Use the arrows to navigate to <Xlist> . If "L1" is not selected, select it. , [L1] , Use the arrows to navigate to <Freq> . Assign the frequencies to [L2] . , [L2] , Go back to access other graphs. , [STAT PLOT] Use the arrows to turn off the remaining plots. Be sure to deselect or clear all equations before graphing. To deselect equations: Access the list of equations. Select each equal sign (=). Continue, until all equations are deselected. To clear equations: Access the list of equations. Use the arrow keys to navigate to the right of each equal sign (=) and clear them. Repeat until all equations are deleted. To draw default histogram: Access the ZOOM menu. Select <9:ZoomStat> . The histogram will show with a window automatically set. To draw custom histogram: Access window mode to set the graph parameters. X min = –2.5 X max = 3.5 X s c l = 1 (width of bars) Y min = 0 Y max = 10 Y s c l = 1 (spacing of tick marks on y -axis) X r e s = 1 Access graphing mode to see the histogram. To draw box plots: Access graphing mode. , [STAT PLOT] Select <1:Plot 1> to access the first graph. Use the arrows to select <ON> and turn on Plot 1. Use the arrows to select the box plot picture and enable it. Use the arrows to navigate to <Xlist> . If "L1" is not selected, select it. , [L1] , Use the arrows to navigate to <Freq> . Indicate that the frequencies are in [L2] . , [L2] , Go back to access other graphs. , [STAT PLOT] Be sure to deselect or clear all equations before graphing using the method mentioned above. View the box plot. , [STAT PLOT] Linear Regression Sample Data The following data is real. The percent of declared ethnic minority students at De Anza College for selected years from 1970–1995 was: Year Student Ethnic Minority Percentage 1970 14.13 1973 12.27 1976 14.08 1979 18.16 1982 27.64 1983 28.72 1986 31.86 1989 33.14 1992 45.37 1995 53.1 The independent variable is "Year," while the independent variable is "Student Ethnic Minority Percent." Student Ethnic Minority Percentage By hand, verify the scatterplot above. NOTE The TI-83 has a built-in linear regression feature, which allows the data to be edited.The x -values will be in [L1] ; the y -values in [L2] . To enter data and do linear regression: ON Turns calculator on. Before accessing this program, be sure to turn off all plots. Access graphing mode. , [STAT PLOT] Turn off all plots. , Round to three decimal places. To do so: Access the mode menu. , [STAT PLOT] Navigate to <Float> and then to the right to <3> . All numbers will be rounded to three decimal places until changed. Enter statistics mode and clear lists [L1] and [L2] , as describe previously. , Enter editing mode to insert values for x and y . , Enter each value. Press to continue. To display the correlation coefficient: Access the catalog. , [CATALOG] Arrow down and select <DiagnosticOn> ... , , r and r 2 will be displayed during regression calculations. Access linear regression. Select the form of y = a + bx . , The display will show: LinReg y = a + bx a = –3176.909 b = 1.617 r = 2 0.924 r = 0.961 This means the Line of Best Fit (Least Squares Line) is: y = –3176.909 + 1.617 x Percent = –3176.909 + 1.617 (year #) The correlation coefficient r = 0.961 To see the scatter plot: Access graphing mode. , [STAT PLOT] Select <1:plot 1> To access plotting - first graph. Navigate and select <ON> to turn on Plot 1. <ON> Navigate to the first picture. Select the scatter plot. Navigate to <Xlist> . If [L1] is not selected, press , [L1] to select it. Confirm that the data values are in [L1] . <ON> Navigate to <Ylist> . Select that the frequencies are in [L2] . , [L2] , Go back to access other graphs. , [STAT PLOT] Use the arrows to turn off the remaining plots. Access window mode to set the graph parameters. X min = 1970 X max = 2000 X s c l = 10 (spacing of tick marks on x -axis) Y min = − 0.05 Y max = 60 Y s c l = 10 (spacing of tick marks on y -axis) X r e s = 1 Be sure to deselect or clear all equations before graphing, using the instructions above. Press the graph button to see the scatter plot. To see the regression graph: Access the equation menu. The regression equation will be put into Y1. Access the vars menu and navigate to <5: Statistics> . , Navigate to <EQ> . <1: RegEQ> contains the regression equation which will be entered in Y1. Press the graphing mode button. The regression line will be superimposed over the scatter plot. To see the residuals and use them to calculate the critical point for an outlier: Access the list. RESID will be an item on the menu. Navigate to it. , [LIST] , <RESID> Confirm twice to view the list of residuals. Use the arrows to select them. , The critical point for an outlier is: 1.9 V SSE n - 2 where: n = number of pairs of data SSE = sum of the squared errors Σ residual 2 Store the residuals in [L3] . , , [L3] , Calculate the Σresidual 2 n - 2 . Note that n - 2 = 8 , [L3] , , , Store this value in [L4] . , , [L4] , Calculate the critical value using the equation above. , , , , , [V] , , [LIST] , , , , [L4] , , , Verify that the calculator displays: 7.642669563. This is the critical value. Compare the absolute value of each residual value in [L3] to 7.64. If the absolute value is greater than 7.64, then the (x, y) corresponding point is an outlier. In this case, none of the points is an outlier. To obtain estimates of y for various x -values: There are various ways to determine estimates for " y. " One way is to substitute values for " x " in the equation. Another way is to use the on the graph of the regression line. TI-83, 83+, 84, 84+ instructions for distributions and tests Distributions Access DISTR (for "Distributions"). For technical assistance, visit the Texas Instruments website at http://www.ti.com and enter your calculator model into the "search" box. Binomial Distribution binompdf( n , p , x ) corresponds to P ( X = x ) binomcdf( n , p , x ) corresponds to P (X ≤ x) To see a list of all probabilities for x : 0, 1, . . . , n , leave off the " x " parameter. Poisson Distribution poissonpdf(λ, x ) corresponds to P ( X = x ) poissoncdf(λ, x ) corresponds to P ( X ≤ x ) Continuous Distributions (general) − ∞ uses the value –1EE99 for left bound ∞ uses the value 1EE99 for right bound Normal Distribution normalpdf( x , μ , σ ) yields a probability density function value (only useful to plot the normal curve, in which case " x " is the variable) normalcdf(left bound, right bound, μ , σ ) corresponds to P (left bound < X < right bound) normalcdf(left bound, right bound) corresponds to P (left bound < Z < right bound) – standard normal invNorm( p , μ , σ ) yields the critical value, k : P ( X < k ) = p invNorm( p ) yields the critical value, k : P ( Z < k ) = p for the standard normal Student's t -Distribution tpdf( x , df ) yields the probability density function value (only useful to plot the student- t curve, in which case " x " is the variable) tcdf(left bound, right bound, df ) corresponds to P (left bound < t < right bound) Chi-square Distribution Χ 2 pdf( x , df ) yields the probability density function value (only useful to plot the chi 2 curve, in which case " x " is the variable) Χ 2 cdf(left bound, right bound, df ) corresponds to P (left bound < Χ 2 < right bound) F Distribution Fpdf( x , dfnum , dfdenom ) yields the probability density function value (only useful to plot the F curve, in which case " x " is the variable) Fcdf(left bound,right bound, dfnum , dfdenom ) corresponds to P (left bound < F < right bound) Tests and Confidence Intervals Access STAT and TESTS . For the confidence intervals and hypothesis tests, you may enter the data into the appropriate lists and press DATA to have the calculator find the sample means and standard deviations. Or, you may enter the sample means and sample standard deviations directly by pressing STAT once in the appropriate tests. Confidence Intervals ZInterval is the confidence interval for mean when σ is known. TInterval is the confidence interval for mean when σ is unknown; s estimates σ. 1-PropZInt is the confidence interval for proportion. NOTE The confidence levels should be given as percents (ex. enter " 95 " or " .95 " for a 95% confidence level). Hypothesis Tests Z-Test is the hypothesis test for single mean when σ is known. T-Test is the hypothesis test for single mean when σ is unknown; s estimates σ. 2-SampZTest is the hypothesis test for two independent means when both σ's are known. 2-SampTTest is the hypothesis test for two independent means when both σ's are unknown. 1-PropZTest is the hypothesis test for single proportion. 2-PropZTest is the hypothesis test for two proportions. Χ 2 -Test is the hypothesis test for independence. Χ 2 GOF-Test is the hypothesis test for goodness-of-fit (TI-84+ only). LinRegTTEST is the hypothesis test for Linear Regression (TI-84+ only). NOTE Input the null hypothesis value in the row below " Inpt ." For a test of a single mean, " μ∅ " represents the null hypothesis. For a test of a single proportion, " p∅ " represents the null hypothesis. Enter the alternate hypothesis on the bottom row.

Math


Tables This module contains links to government site tables used in statistics. Tables (NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, January 3, 2009) Critical Values of the Student's t Distribution Cumulative Distribution Function of the Standard Normal Distribution Critical Values of the Chi-Square Distribution Upper Critical Values of the F Distribution Tables for Probability Distributions

Math