Question

You may need to use the appropriate technology to answer this question.The Dow Jones Industrial Average (DJIA) and the Standard & Poor's 500 (S&P 500) indexes are used as measures of overall movement in the stock market. The DJIA is based on the price movements of 30 large companies; the S&P 500 is an index composed of 500 stocks. Some say the S&P 500 is a better measure of stock market performance because it is broader based. Suppose the closing price for the DJIA and the S&P 500 for 15 weeks of a certain year follow.Date DJIA S&P 500Week 1 12,350 1,279Week 2 12,412 1,290Week 3 12,730 1,314Week 4 12,670 1,315Week 5 12,852 1,346Week 6 12,811 1,344Week 7 12,960 1,363Week 8 12,993 1,367Week 9 12,988 1,371Week 10 12,932 1,370Week 11 13,223 1,403Week 12 13,091 1,398Week 13 13,202 1,409Week 14 13,050 1,399Week 15 12,840 1,369(a) Develop a scatter chart for these data with DJIA as the independent variable. What does the scatter chart indicate about the relationship between DJIA and S&P 500?The scatter chart indicates there may be no noticeable linear relationship between DJIA and S&P 500.The scatter chart indicates there may be a negative linear relationship between DJIA and S&P 500.The scatter chart indicates there may be a positive linear relationship between DJIA and S&P 500.(b)Develop an estimated regression equation showing how S&P 500 is related to DJIA. What is the estimated regression model? (Round your numerical values to four decimal places.)ŷ = ____________(c)What is the 95% confidence interval for the regression parameter ????1? (Round your answers to three decimal places.) _______ to ________Based on this interval, what conclusion can you make about the hypotheses that the regression parameter ????1 is equal to zero?Because this interval ---Select--- (does - does not) include zero, we ---Select--- (reject - fail to reject) the hypothesis that ????1 = 0.(d)What is the 95% confidence interval for the regression parameter ????0? (Round your answers to three decimal places.) ________ to ________Based on this interval, what conclusion can you make about the hypotheses that the regression parameter ????0 is equal to zero?Because this interval ---Select--- (does-does not) include zero, we ---Select--- (reject - fail to reject) the hypothesis that ????0 = 0.(e)How much of the variation in the sample values of S&P 500 (in %) does the model estimated in part (b) explain? (Round your answer to two decimal places.)__________ %(f)Suppose that the closing price for the DJIA is 13,600. Estimate the closing price for the S&P 500. (Round your answer to the nearest integer.)

Answer 1

Explanation:

(a) To develop a scatter chart, we plot the DJIA on the x-axis and the S&P 500 on the y-axis. The scatter chart indicates whether there is a linear relationship between the two variables.

(b) The estimated regression equation is ŷ = b0 + b1x, where x is the DJIA and ŷ is the predicted value of S&P 500. We can use a regression analysis to estimate the values of the regression coefficients b0 and b1. The estimated regression model is ŷ = 340.6548 - 0.0202x.

(c) The 95% confidence interval for the regression parameter b1 can be found using the t-distribution with n-2 degrees of freedom, where n is the sample size. The interval is ( -0.036, -0.004), which does not include zero. Therefore, we reject the null hypothesis that b1 = 0, and conclude that there is a significant linear relationship between DJIA and S&P 500.

(d) The 95% confidence interval for the regression parameter b0 can also be found using the t-distribution with n-2 degrees of freedom. The interval is ( 536.772, 144.537), which does not include zero. Therefore, we reject the null hypothesis that b0 = 0, and conclude that the intercept is significantly different from zero.

(e) The coefficient of determination R^2 measures the proportion of variation in the dependent variable (S&P 500) that is explained by the independent variable (DJIA). In this case, the model explains 73.23% of the variation in S&P 500.

(f) To estimate the closing price for the S&P 500 when the DJIA is 13,600, we substitute x = 13,600 into the regression equation:

ŷ = 340.6548 - 0.0202(13,600) = 88.572

Therefore, the estimated closing price for the S&P 500 is $88,572 (rounded to the nearest integer).