[1] A real estate agency collects data concerning y= the sales price of a house (in thousands of dollars), and x = the home size (in hundreds of square feet)

1. Find the value of the linear correlation coefficient r.
2. Find the value of the coefficient of determination r ² , and interpret the meaning for this problem.
3. Is there a linear correlation between home size and sales price?  Test it at the 0.05 significance level.
4. If there is a linear correlation, what is the regression equation?
5. Interpret the meaning of the slope b ₁ in this problem.
6. Interpret the meaning of the y-intercept b ₀ in this problem.  Will this make sense to the data set?
7. Using (d), find the predicted sales price that has home size of 1200 ft ² .
8. Find the standard error of estimate S _YX .
9. Find the total variation SST.
10. Using (b) and (i), find the explained variation SSR.

[2] U.S. banks have been merging to form mega banks that span many states.  The table given lists the number of U.S. bank mergers each year from 1980 (year 1) to 1993 (year 14) for which $50 million or more changed hands in the transaction.

1. Assuming a linear relationship between x and y, find the regression coefficients b ₀ and b ₁ .
2. Interpret the meaning of b ₁ in this problem.
3. Find the correlation coefficient r.
4. At the 0.05 level of significance, is there a significant correlation between x and y?
5. Compute the coefficient of determination r ² and interpret its meaning for the data.
6. Based on results from (c) to (e), how many mergers would be predicted in 1994 (year 15)?  Compare your answer to the actual number of 1994 mergers: 42.

[3] The following data represents the digital-mode talk time in hours and the battery capacity in milliampere-hours of cell phones.

1. Compute the correlation coefficient r.
2. At the 0.05 level of significance, is there a significant correlation between the battery capacity and the digital-mode talk time?
3. What conclusions can you reach about the relationship between the battery capacity and the digital-model talk time?
4. You would expect the cell phones with higher battery capacity to have a higher talk time.  Is this borne out by the data?

[4] Each year ninth-grade students in Ohio must take a proficiency test.  The data (SCHOOL.xls in the CD) contains data from 47 school districts in Ohio from 1994 to 1995 school year.  The variables in the data set are:

– School District: Name of school district

– Percentage Passing: Percentage of students passing the ninth-grade proficiency test

– Percentage Attendance: Daily average of the percentage of students attending class

– Salary: Average teachers salary (dollars)

– Spending: Instructional spending per pupil (dollars)

1. Set up a scatter diagram using the percentage passing the proficiency test as the dependent variable and daily attendance as the independent variable.  Discuss the scatter diagram.
2. Assuming a linear relationship, find the regression coefficients, b ₀ , b ₁ , and its regression equation.
3. Interpret the meaning of the slope b ₁ in this problem.
4. Find the standard error of the estimate.
5. Determine the coefficient of determination, r ² , and interpret its meaning.
6. Compute the coefficient of correlation r and interpret its meaning.
7. Perform a residual analysis on your results and determine the adequacy of the fit of the model.
8. At the 0.05 level of significance, is there evidence of a linear relationship between the independent variable and the dependent variable?
9. Set up a 95% confidence interval estimate of the population slope, b ₁ and interpret its meaning.
10. Repeat (a)–(i) using instructional spending as the independent variable.

1. Set up a 95% confidence interval estimate of the population slope, b ₁ and interpret its meaning.

k)       Which of the two models is best at predicting the percentage of students who will pass the ninth-grade proficiency test?  Write a short summary of your finding.

[5] In Problem [4], simple linear regression models were constructed to investigate the relationships between passing rate of the Ohio ninth-grade proficiency exam and two different independent variables.  Develop the most appropriate multiple regression model to predict a school's passing rate.  Be sure to perform a thorough residual analysis.  In addition, provide a detailed explanation of the results, including a comparison of the most appropriate multiple regression model to the best simple linear regression model.

[6] A collector of antique grandfather clocks believes that the price (in dollars) received for the clocks at an antique auction increases with the age of the clocks and with the number of bidders.  Thus the model is hypothesized is Y = b ₀ + b ₁ x ₁ + b ₂ x ₂ + e, where y = auction price, x ₁ = age of clock (years) and x ₂ = number of bidders.

A sample of 32 auction prices of grandfather clocks, along with their ages and the number of bidders, is given in the data (AUCTION.xls).

1. State the multiple regression equation.
2. Interpret the meaning of the slopes b ₁ and b ₂ in the model.
3. Interpret the meaning of the regression coefficient b ₀ .
4. Test H ₀ : b ₂ = 0 against H ₁ : b ₂ > 0.  Interpret your finding.
5. Use a 95% confidence interval to estimate b ₁ .  Interpret the p-value corresponding to the estimate b ₁ .  Does the confidence interval support your interpretation in b)?
6. Determine the coefficient of multiple determination r ² _Y.12 and interpret its meaning.
7. Perform a residual analysis on your results and determine the adequacy of the fit of the model.
8. Plot the residuals against the prices.  Is there evidence of a pattern in the residuals?  Explain.
9. At a = 0.05, is there evidence of positive autocorrelation in the residuals?
10. Suppose the collector, having observed many auctions, believes that the rate of increase of the auction price with age will be driven upward by a large number of bidders.  In other words, the collector believes that the age of clock and the number of bidders should interact.  Is there evidence to support his claim that the rate of change in the mean price of the clocks with age increases as the number of bidders increases?  Should the interaction term be included in the model?  If so, what is the multiple regression equation?

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Solution: The downloadable solution consists of 27 pages, 2539 words and 14 charts.
Deliverable: Word Document