6A Total Marks = 63 Please show all your work. Some questions require you to use graphs, tree diagrams, formulas, and so on that are difficult or impossible to reproduce in word-processing programs, so you will have to print the entire assignment and fill it out and mail it to your tutor. As an alternative to mailing, you might be able to photograph or scan your assignment and submit it through the appropriate drop box. If you do, make sure the material you are submitting is legible. Question 1 A manufacturer of exercise bicycles is studying the relationship between the number of months an exercise bicycle has been owned and the length of time it was used last week. Data for a random sample of 7 owners are provided in the table below Months Owned x 14 6 8 8 5 7 3 9 10 6 4 8 12 Introduction to Statistics Hours Used Last Week y 7 1 4 marks a. Construct a scatter diagram for these data with \"Months Owned\" on the horizontal axis, and \"Hours Used Last x Week\" on the vertical y axis. Note: Try to make relatively full use of the graph area. 2 marks b. Describe the general pattern of the relationship between the two variables. 11 marks c. Calculate the least squares regression line with hours used as the dependent variable and months owned as the independent variable. Note: Express a and b to 4 decimal places of accuracy. 3 marks d. Plot the regression line on the scatter diagram you constructed in part (a). Note: Show your calculations for 2 x4 and x1 2 . Mathematics 215 / Assignment 6A 10 marks e. Can it be concluded that the slope of the regression line is negative? Formulate and test the appropriate hypotheses at the 5% significance level. Use the critical value approach. Introduction to Statistics 3 4 marks f. Construct a 95% confidence interval for B. 2 marks g. Interpret the numerical value of b in the sample regression line. What does it mean in the context of this question? 2 marks h. Use the equation of the regression line to predict hours used last week if an exercise bicycle has been owned for 11 months. Note: Express your answer to 2 decimal places of accuracy. 4 Mathematics 215 / Assignment 6A Question 2 An ice cream vendor would like to predict daily sales of ice cream cones based on the temperature outdoors. Data for a random sample of 8 days are provided in the table below. Temperature (C) x Daily Sales (Litres) y 6 135 8 50 12 125 16 100 18 200 22 150 28 250 28 175 The regression line with \"daily sales\" as the dependent variable and \"temperature\" as the independent variable has been calculated as: y 55.217 5.38597x You may use the following sums of squares and cross products for the questions below. x 138 y 1185 SSxx 495.5 SSyy 26446.875 SSxy 2668.75 2 marks a. Interpret the value of b in the sample regression line. What does it mean in the context of this question? 2 marks b. Compute the linear correlation between \"daily sales\" and \"temperature.\" Note: Express your answer to 4 decimal places of accuracy. Introduction to Statistics 5 8 marks 2 marks 6 c. At the 5% significance level, can it be concluded that the correlation between daily sales and temperature is positive? Formulate and test the appropriate hypotheses. Use the critical value approach. d. What percentage of the variation in daily sales is explained by its linear relationship with temperature? Mathematics 215 / Assignment 6A 11 marks e. Using the regression line and sums of squares and cross products provided above, construct a 90% prediction interval for daily sales when the outside temperature is 24C. Introduction to Statistics 7