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## Regression Assignment: What is the probability of rolling a pair of dice and obtaining a total score of 9 or more?

Chapter 5: Probability 11. ——— 5.1. (a) What is the probability of rolling a pair of dice and obtaining a total score of 9 or more? (b) What is the probability of rolling a pair of dice and obtaining a total score of 7? 12. ——— 5.2. A box contains four black pieces of cloth, two striped pieces, and six dotted pieces. A piece is selected randomly and then placed back in the box. A second piece is selected randomly. What is the probability that: a. both pieces are dotted? b. the first piece is black and the second piece is dotted? c. one piece is black and one piece is striped? Chapter 7: Normal Distribution 13. ——— 7.1. If scores are normally distributed with a mean of 35 and a standard deviation of 10, what percent of the scores is: a. greater than 34? b. smaller than 42? c. between 28 and 34? 14. ——— 7.2. What are the mean and standard deviation of the standard normal distribution? (b) What would be the mean and standard deviation of a distribution created by multiplying the standard normal distribution by 8 and then adding 75? 15. ——— 7.4. What proportion of a normal distribution is within one standard deviation of the mean? (b) What proportion is more than 2.0 standard deviations from the mean? (c) What proportion is between 1.25 and 2.1 standard deviations above the mean? 16. ——— 7.5. A test is normally distributed with a mean of 70 and a standard deviation of 8. (a) What score would be needed to be in the 85th percentile? (b) What score would be needed to be in the 22nd percentile? 3 of 6 17. ——— 7.11. A group of students at a school takes a history test. The distribution is normal with a mean of 25, and a standard deviation of 4. (a) Everyone who scores in the top 30% of the distribution gets a certificate. What is the lowest score someone can get and still earn a certificate? (b) The top 5% of the scores get to compete in a statewide history contest. What is the lowest score someone can get and still go onto compete with the rest of the state? Chapter 14 – Regression

18. ——— 14.2. The formula for a regression equation is Y’ = 2X + 9. a. What would be the predicted score for a person scoring 6 on X? b. If someone’s predicted score was 14, what was this person’s score on X? 19. ——— 14.6. For the X,Y data below, compute: b. the slope of the regression line and test if it differs significantly from zero. c. the 95% confidence interval for the slope. 20. ——— 14.10. Using linear regression, find the predicted post-test score for someone with a score of 45 on the pre-test. X Y 4 6 3 7 5 12 11 17 10 9 14 21 4 of 6 Pre Post 42 48 41 58 45 36 27 13 63 50 Pre Post 54 81 44 56 50 64 47 50 55 63 Pre Post 59 56 52 63 44 55 51 50 42 66 Pre Post 49 57 45 73 57 63 46 46 60 60 Pre Post 65 47 64 73 50 58 74 85 59 44 From Penn State STAT501 https://onlinecourses.science.psu.edu/stat501/node/254/ 21. ——— From 1.4 The equation W = -266.5 + 6.138H is the regression equation representing the relationship between height in inches and weight in pounds for some group of people, whose measurements are shown in the table. (a) Show that the root mean square error is 8.64137. You may do this with a spreadsheet or as an original computer program, but I want to see the general equations and calculated numbers for – the predicted weight for each height – the residual for each individual’s actual measurements – each element of the mean square error – the mean square error (variance, σ2) – the standard deviation (standard deviation of the mean, root mean square error, σ) (b) Create one or more plots to explore the residuals. What do you observe? (c) Why do we usually choose to report σ rather than σ2?

22. ——— {See Section 5.4} For the same data above, use the matrix formulation to show that W = -266.5 + 6.138H. {See Section 5.4} (a) Write the basic matrix equation with H and W. Be careful about which is used in the X position in which in the Y position. (b) Plug matrices in for X, XT, and Y (c) Find XTX and XXT. (d) (XTX)-1. (You may use a computer!) (e) Find XTY (f) Solve the equation. (g) Show how to translate your solution to the regression equation. 23. ——— From 13.1. Consider the famous 1877 Galton dataset, consisting of 7 measurements each of X = Parent (pea diameter in inches of parent plant) and Y = Progeny (average pea diameter in inches of up to 10 plants grown from seeds of the parent plant). Also included in the dataset are standard deviations, σ, of the offspring peas grown from each parent. These standard deviations 5 of 6 Height (H) Weight (W) 63 127 64 121 66 142 69 157 69 162 71 156 71 169 72 165 73 181 75 208 reflect the information in the response Y values (remember these are averages) and so in estimating a regression model we should downweight the obervations with a large standard deviation and upweight the observations with a small standard deviation. In other words we should use weighted least squares with weights equal to 1/σ2. The resulting fitted equation is: Progeny = 0.12796 + 0.2048 Parent (a) Assume that 65 parent plants are measured to have diameters of 0.21in. How many progeny plants do you expect do find with diameters in the range 0.15272-0.19248in? (b) Set up the matrix equation with XT, W, X, and Y matrices. (c) Find WX (d) Find XTWX (e) Find (XTWX)-1 (f) Find WY (g) Find XTWY (h) Solve the full matrix equation (i) Translate your solution into the regression equation