F2023STAT213.30233251.Assignment_6 (1)

Shubkarman Gill F2023STAT213 Assignment 6 is due on Friday, November 24, 2023 at 11:59pm. The number of attempts available for each question is noted beside the question. If you are having trouble figuring out your error, you should consult the textbook, or ask a fellow student, one of the TA’s or your professor for help. There are also other resources at your disposal, such as the Mathematics Continuous Tutorials. Don’t spend a lot of time guessing – it’s not very efficient or effective. Make sure to give lots of significant digits for (floating point) numerical answers. For most problems when entering numerical answers, you can if you wish enter elementary expressions such as 2 ∧ 3 instead of 8, sin ( 3 * pi / 2 ) instead of -1, e ∧ ( ln ( 2 )) instead of 2, ( 2 + tan ( 3 )) * ( 4 - sin ( 5 )) ∧ 6 - 7 / 8 instead of 27620.3413, etc. Problem 1. (1 point) Match the correlation coefficients with their scatterplots. Select the letter of the scatterplot below which corresponds to the correlation coefficient. (Click on image for a larger view.) ? 1. r = - 0 . 74 ? 2. r = 0 . 89 ? 3. r = - 0 . 97 ? 4. r = 0 . 76 A B C D Answer(s) submitted: • A • D • B • C submitted: (correct) recorded: (correct) Correct Answers: • A • D • B • C 1

Problem 2. (1 point) The amounts of 6 restaurant bills and the corresponding amounts of the tips are given in the below. Assume that bill amount is the explanatory variable and tip amount the response variable. Bill 32 . 98 97 . 34 49 . 72 52 . 44 43 . 58 70 . 29 Tip 4 . 50 16 . 00 5 . 28 7 . 00 5 . 50 10 . 00 (a) Find the correlation: r = (b) Does there appear to be a significant correlation? • A. No • B. Yes (c) The regression equation is ˆ y = . (d) If the amount of the bill is $85 , the best prediction for the amount of the tip is $ . Note: Enter your answer as a number xx.xx (e) According to the regression equation, for every $5 increase in the bill, the tip should (Enter INCREASE or DECREASE) by $ . Answer(s) submitted: • 0 . 98274 • B • - 2 . 7252 + 0 . 1866 x • 13 . 14 • INCREASE • 0 . 933 submitted: (correct) recorded: (correct) Correct Answers: • 0 . 98274 • B • - 2 . 72516 + 0 . 186606 x • 13 . 1363 • INCREASE • 0 . 93303 2

Problem 4. (1 point) The following data represents the winning percentage (the number of wins out of 162 games in a season) as well as the teams Earned Run Average, or ERA. The ERA is a pitching statistic. The lower the ERA, the less runs an opponent will score per game. Smaller ERA’s reflect (i) a good pitching staff and (ii) a good team defense. You are to investigate the relationship between a team’s winning percentage - Y , and its Earned Run Average (ERA) - X . Winning Proportion - Y Earned Run Average (ERA) - X 0.623457 3.13 0.512346 3.97 0.635802 3.68 0.604938 3.92 0.518519 4.00 0.580247 4.12 0.413580 4.29 0.407407 4.62 0.462963 3.89 0.450617 5.20 0.487654 4.36 0.456790 4.91 0.574047 3.75 (a) Using R-Studio, create a scatter-plot of the data. What can you conclude from this scatter-plot? • A. There is not a linear relationship between the a teams winning percentage and its ERA. • B. There is a positive linear relationship between a teams winning percentage and its ERA. • C. There is a negative linear relationship between a teams winning percentage and its ERA. (b) Use R-Studio to find the least squares estimate of the linear model that expressed a teams winning percentage as a linear function of is ERA. Use four decimals in each of your answers. b Y i = [?/+/-] X i (c) Find the value of the coefficient of determination, then complete its interpretation. r 2 = (use four decimals) The percentage of 4

• ? • variation • standard deviation • the mean in • ? • a teams winning percentage • a teams earned run average that is explained by its linear relationship with • ? • the teams winning percentage • the teams earned run average is %. (d) Interpret the meaning of the slope term in the estimate of the linear model, in the context of the data. As a teams • ? • winning percentage • earned run average increases by • ? • one percentage point • one earned run the teams • ? • winning percentage • earned run average will • ? • will increase by an average of • will decrease by an average of • will increase by • will decrease by . (use four decimals) (e) A certain professional baseball team had an earned run average of 3.45 this past season. How many games out of 162 would you expect this team to win? Use two decimals in your answer. games won (f) The team mentioned in part (e) won 91 out of 162 games. Find the residual, using two decimals in your answer. 5

Problem 5. (1 point) Studies suggest that women who smoke during pregnancy affect the birth weights of their newborn infants. To study this issue further, a sample of 10 women smokers were asked to estimate the average number of cigarettes they smoke per day. In the following table are their estimates and the birth weight (in kgs) of their child. Average Number of Cigarettes per day Birth Weight of Child (in kilograms) 22 2.91 16 3.27 4 3.68 19 3.14 42 2.32 8 3.82 12 3.45 30 2.95 14 3.00 16 3.68 (a) Using R-Studio, create a scatter-plot of the data. What can you conclude from this scatter-plot? • A. There is a positive linear relationship between the average number of cigarettes the mother con- sumes and the birthweight of her child. • B. There is a negative linear relationship between the average number of cigarettes the mother consumes and the birthweight of her child. • C. There is not a linear relationship between the average number of cigarettes the mother consumes and the birthweight of her child. (b) Use R-Studio to find the least squares estimate of the linear model that expresses the birth weight of a child as a linear function of the average number of cigarettes mom consumes in a day. Use four decimals in each of your answers. b Y i = [?/+/-] X i (c) Find the value of the coefficient of determination, then complete its interpretation. r 2 = (use four decimals) The percentage of • ? • variation • standard deviation • the mean 7

in • ? • the birthweight of the child • the average number of cigarettes consumed in a day by the mother that is explained by its linear relationship with • ? • the birthweight of the child • the average number of cigarettes consumed in a day by the mother is %. (d) Interpret the meaning of the slope term in the estimate of the linear model, in the context of the data. As • ? • the number of cigarettes consumed in a day • birthweight increases by • ? • by one on average • by one kilogram the • ? • birthweight • average number of cigarettes consumed will • ? • will increase by an average of • will decrease by an average of • will increase by • will decrease by . (use four decimals) (e) A expecting mother, who smokes, claims to smoke an average of 20 cigarettes in a day. Predict the weight of her newborn. Use two decimals in your answer. kilograms (f) The weight of the child born to the expecting mother mentioned in (e) was 4.2 kilograms. Find the residual, use two decimals. e i = Answer(s) submitted: 8

Problem 6. (1 point) The Capital Asset Price Model (CAPM) is a financial model that attempts to predict the rate of return on a financial instrument, such as a common stock, in such a way that it is linearly related to the rate of return on the overal market. Specifically, R StockA , i = β 0 + β 1 R Market , i + e i You are to study the relationship between the two variables and estimate the above model: R StockA , i - rate of return on Stock A for month i , i = 1 , 2 , ··· , 59. R Market , i - market rate of return for month i , i = 1 , 2 , ··· , 59. β 1 represent’s the stocks ‘beta’ value, or its systematic risk. It measure’s the stocks volatility related to the market volatility. β 0 represents the risk-free interest rate. The data in the Download .csv file contains the data on the rate of return of a large energy company which will be referred to as Acme Oil and Gas and the corresponding rate of return on the Toronto Composite Index (TSE) for 59 randomly selected months. Therefore R Acme , i represents the monthly rate of return for a common share of Acme Oil and Gas stock; R TSE , i represents the monthly rate of return (increase or decrease) of the TSE Index for the same month, month i . The first column in this data file contains the monthly rate of return on Acme Oil and gas stock; the second column contains the monthly rate of return on the TSE index for the same month. (a) Using R-Studio, create a scatter-plot of the data. What can you conclude from this scatter-plot? • A. There is a negative linear relationship between the monthly rate of return of Acme stock and the monthly rate of return of the TSE Index. • B. There is not a linear relationship between the monthly rate of return of Acme stock and the monthly rate of return of the TSE Index. • C. There is a positive linear relationship between the monthly rate of return of Acme Oil and Gas stock and the monthly rate of return of the TSE Index. (b) Find the value of the correlation coefficient. Enter your answer using four decimals. r = (c) Use R-Studio to find the least squares estimate of the linear model that expressed the monthly rate of return on Acme Oil and Gas stock as a linear functino of the monthly rate of return on the Toronto Stock Exchange Index. Use three decimals in each of your answers. 10

\ R Acme , i = + R TSE , i (d) The rate of return on the TSE Index for the month of October was -3.42%. Predict the monthly rate of return of Acme Oil and Gas stock for the month of October. Use four decimals in your answer. b R Acme , Oct = % (e) Interpret the meaning of the predicted value in (d) in the context of the data. If the monthly rate of return of the TSE Index is -3.42 %, the • ? • monthly rate of return • normal monthly rate of return • mean monthly rate of return of Acme Oil and Gas stock is . (f) Find the coefficient of determination. Expresses as a percentage, and use two decimal places in your answer. r 2 = % (g) In the context of the data, interpret the meaning of the coefficient of determination. • A. There is a strong, positive linear relationship between the monthly rate of return of Acme stock and the monthly rate of return of the TSE Index. • B. There is a weak, positive linear relationship between the monthly rate of return of Acme stock and the monthly rate of return of the TSE Index. • C. The percentage found above is the percentage of variation in the monthly rate of return of Acme stock that can be explained by its linear relationship with the monthly rate of return of the TSE Index. • D. The percentage found above is the percentage of variation in the monthly rate of return of the TSE Index that can be explained by its linear dependency with the monthly rate of return of Acme stock. (h) Find the residual corresponding to the data point found in the 10th row of the data file. Use four deci- mals in your answer. e 10 = 11