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Homework 10
Problem 1
Let the sample regression line (least square regression line) between the predictor x and the response y be y^ = 50 − 3x. The size of the sample collected to obtain this regression line was 30.
Part I
If the square of the correlation coefficient is 0.81, then which of the following options is correct?
1. r = 0.9
2. r = −0.81
3. r = −0.9
4. r = 0.81
Part II
If the sample mean yˉ = −49 what is the value of xˉ?
1. xˉ = 33
2. xˉ = 49
3. xˉ = 50
4. xˉ = −49
Part III
If we want to test H0 : β = 0 agains Ha : β > 0 at 1% significance level, what can we say about the p-value?
1. p-value >0.01
2. p-value = 0.5
3. p-value >0.54. p-value < 0.01
Part IV
If the p-value for testing H0 : β = 0 against Ha : β = 0 is 0.34. Which of the following R-code would give us the value of the test statistic?
1. qt(-0.17, df = 28)
2. −qt(0.17, df = 28)
3. −qt(0.17, df = 28, lower = FALSE)
4. qt(0.34, df = 28, lower = FALSE)
Problem 2
Suppose you want to study the association between the revenue and the total CEO compensation of 23 large companies using the dataset Company.csv.
The dataset has the following two columns:
TotalCompensation: The revenue in $1000 dollars of 23 large companies
Revenue : the total CEO compensation in $1000 for these companies
Use the following R command and output to answer numerical questionsLet μY = α + βx be the population regression equation.
1. Identify the explanatory variable.
2. Write down the estimated regression equation.
3. Interpret the slope and the y-intercept in context. Comment whether this interpretation of the intercept is realistic or not.
4. What is the mean/predicted total CEO compensation if the revenue of the company is 1,000,000 unit revenue?
5. Identify r
2
and calculate r using the value of r
2
. Give an interpretation of the strength and the direction of r.
6. Would you recommend using the least squares regression line obtained in part (ii) to predict the CEO’s total compensation given a company revenue of 500,000? Why?
Hint: You can compute the maximum, using the max() command and mini- mum,using the min() command, values of Revenue. The data set is the .csv file (Company.csv) posted on Canvas under the module week 13. Download and import ,the file into R using the following code. You have to insert your own pathname. Remember to type the following code into R instead of cut- ting/pasting it.
7. Compute the correlation coefficient by using R since you have access to the data.
8. Find the equation of the sample regression line without using the above R output.
Problem 3
Suppose the dataset ReturnSPRTX1 contains 46 observations and two variables:
SP500: return of the S&P 500 stock
RTX: return of the Raytheon Technologies Corporation
Use the following R output to answer questions. (The dataset is not provided. You don’t need it.)
Assume that the conditions of the linear regression are met.
1. Based on the information provided in the R output, which variable is the response variable?
There is no need to explain.
2. Based on the information provided in the R output, construct a 98% confidence interval for the slope in the population regression equation. Show your work.
An expert claimed that the return of the Raytheon Technologies Corporation is less than 0.85 times the return of SP500. Evaluate whether the expert’s claim is supported by our data using an hypothesis test at 5% significance level by answering Part 3-6.
3. Write the null and alternative hypotheses in statistical notation. Hint: the alternative ,hypothesis is one-sided.
4. Calculate the test statistic. Please show work.
5. Calculate the p-value. Please show work. Hint: your p-value should be greater than 0.25. Otherwise something is wrong.
6. Does our dataset provide sufficient evidence for the expert’s claim? Explain.