1. A regression model relating x, number of sales persons at a branch office, to y, annual sales at the office ($1000s), has been developed. The computer output from a regression analysis of the data follows.
   The regression equation is
   = 80.0 + 50.0X
   Predictor Coef Stdev t-ratio
   Constant 80.0 11.333 7.06
   X 50.0 5.482 9.12
   Analysis of Variance
   SOURCE DF SS MS
   Regression 1 6828.6 6828.6
   Error 28 2298.8 82.1
   Total 29 9127.4
   a. Write the estimated regression equation
   b. How many branch offices were involved in the study?
   c. Compute the F statistic and test the significance of the relationship at a .05 level of significance.
   d. Predict the annual sales at the Memphis branch office. This branch has 12 sales persons.
2. Find the following:
   
   (a) Compute the sample regression coefficients b _o and b ₁ .
   (b) Compute the estimated variance of the regression.
   (c) Compute the standard error of the regression.
   (d) Compute the estimated variance of b ₁ .
   (e) Compute the standard error of b ₁ .

   Year	y = book value per share	x = earning per share
   1980	$ 0.15	$0.04
   1981	0.17	0.02
   1982	0.28	0.11
   1983	0.38	0.10
   1984	0.51	0.13
   1985	0.81	0.35
   1986	1.32	0.57
   1987	1.87	0.65
   1988	2.48	0.70
   1989	2.88	0.50

3. Answer the following questions:

1. If r ² = 0.95, n = 11 and the \[\sum{{{\left( y-\bar{y} \right)}^{2}}}\] = 100, what is \[S_{e}^{2}\] or \[S_{y\backslash x}^{2}\] ?
2. If r ² = 1, then \[S_{e}^{2}\] in (1a) can be shown to have what type of relationship with SST and SSR?
3. What relationship exists between \[\hat{y}\] and Y where r ² = 1?
4. What relationship exists between \[\hat{y}\] and Y where r ² = 0?
5. Given the following information: r ² = 0.95, n = 10 and k =2 what is the t-value for b ₁ ?
6. In (e), you calculated t-value at the 0.05 level of significance; what statistical decision can you make about b ₁ ?

1. Calculate r ² for the following sets of data:

1. \[\sum{{{\left( y-\bar{y} \right)}^{2}}}=210\] and \[\sum{{{\left( \hat{y}-\bar{y} \right)}^{2}}}\] = 140
2. \[\sum{{{\left( y-\bar{y} \right)}^{2}}}=100\] and \[\sum{{{\left( y-\hat{y} \right)}^{2}}}\] = 10
3. \[\sum{{{\left( \hat{y}-\bar{y} \right)}^{2}}}=75\] and \[\sum{{{\left( y-\hat{y} \right)}^{2}}}\] = 25

1. Answer the following questions:

1. Given the equation: \[\hat{y}\] = 10 + 0.69x ₁ ; n =10 and r ² = 0.96, write down the p-value for your calculated t-statistic.
2. Calculate the coefficient of correlation from the information in (a). Write the H _o and H ₁ . Then, make a decision about the significance of the coefficient of correlation.
3. What is the \[S_{{{b}_{1}}}^{2}\] in (a)? If SST = 20 in (a), what is SSE? What is SSR?

1. Given the regression equation: \[\hat{Y}\] = 1.3479 + 0.3978 X, what is the fitted value (or \[\hat{Y}\] ) if X = -3?
2. Calculate b _o , b ₁ , \[S_{{{b}_{1}}}^{2}\] , and \[{{S}_{e}}\] for the information provide below

   X	Y
   -2	9
   0	5
   -0.5	7
   1	100

   
   Explain the theoretical meaning of b _o , b ₁ , \[S_{{{b}_{1}}}^{2}\] , and \[{{S}_{e}}\] .
3. Four different assembly processes were under consideration. Sixteen workers were randomly assigned to the four processes, eight per process. The number of correctly assembled units in an eight-hour work shift was recorded:

   Process 1	Process 2	Process 3	Process 4
   31	29	28	32
   36	32	36	33
   36	35	29	33
   34	32	31	31

   What is the value of SS F

   b. What is the value of SS T

   c. What is the value of SS E

   d. With the \[\alpha \] = 0.05, is there a significant difference between the four process?

4. The following regression equation was obtained using the five independent variables.

1. What percent of the variation is explained by the regression equation?
2. What is the standard error of regression ?
3. What is the critical value of the F-statistic ?
4. What sample size is used in the print out?
5. What is the variance of the slope coefficient of income?
6. Conduct a global test of hypothesis to determine if any of the regression coefficients are not zero.
7. Conduct a test of hypothesis on each of the independent variables. Would you consider eliminating outlets and bosses?

1. The commercial division of a real estate firm is conducting a regression analysis of the relationship between gross rents (X) and selling price (Y) for apartment building. Data have been collected on several properties recently sold, and the following output has been obtained in computer run.
   

   Question

   How many apartment buildings were in the sample

   b. Write the estimated regression equation

   c. What’s the sample regression coefficients of b 0

   d. What’s the sample regression coefficients of b 1

   e. What’s the value of the estimated variance of the regression

   f. What’s the value of the standard error of the regression

   g. What’s the value of the estimated variance of the b 1

   h. What’s the value of the standard error of the b 1

   Is the result significant when using F statistic to test the significant of the relationship at a 0.05 level of significance.

   j. Estimate the selling price of an apartment building with gross annual rent of $50,000

2. A metropolitan bus system sampler’s rider counts on one of its express commuter routes for a week. Use the following data to establish whether rider ship is evenly balanced by day of the week. Let \[\alpha \] = 0.05.

1. Is the ² value significant at 5% level of significant?
2. Write the conclusion for this question

1. Test each of the following hypotheses:

1. H _o : β = 0
   H _a : β ≠ 0
   \[{{\hat{\beta }}_{1}}=1.87;{{S}_{{{{\hat{\beta }}}_{1}}}}=0.63;n=3;\alpha =0.05\]
2. H _o : β = 0
   H _a : β < 0
   \[{{\hat{\beta }}_{1}}=-26.32;{{S}_{{{{\hat{\beta }}}_{1}}}}=14.51;n=100;\alpha =0.01\]
3. H _o : β = 0

H _a : β > 0

1. a) If \[\hat{Y}\] = 1.3479 + 0.3978x, what is the x value? ( \[\hat{Y}\] = -3)
   b) True or False. Regression Analysis attempts to minimize the mean square error.
2. Computer output:

ANOVA

S _e =12.86986 R-sq = 0.862263 R-sq(adj) = 0.8393068

1. Write and interpret the multiple regression equation.
2. Does the model with Price and Advertising contribute to the prediction of Y? Use a 0.05 significance level.
3. Which independent variable appears to be the best predictor of sales? Explain.
4. What is the number of observations used in this study?
5. Assuming that the coefficient on Advertising has H _a : B1 > 0, what statistical decision should be made at 5% level.
6. What is the standard error of estimate? Can you use this statistic to assess the model’s fit? If so, how?
7. What is the coefficient of determination, and what does it tell you about the regression model?
8. What is the coefficient of determination, adjusted for degrees of freedom? What do this statistic and the statistic referred to in part (g) tell you about how well this model fits that data.
9. Test the overall utility of the model. What does the p-value of the test statistic tell you?

1. The Acme corp. wished to predict maintenance costs per year on its factory equipment by knowledge of the equipment’s age. The following data was collected which yielded the regression equation: (see the printout)

1. What is the point estimate of maintenance cost for a machine that is 3 years old?
2. Determine the 4 missing values in the printout given below.

1. Using the following Y and X

1. What is regression analysis?
2. Write the regression equation.
3. Interpret the meaning of slope coefficient.
4. What is another name of r ² since r ² is the symbol?
5. What is another name of r since r is the symbol?
6. Name of 4 parts of the regression equation.
7. If r is 0.89, what is r ² what is the expected sign of slope coefficient.
8. Calculate r ² using the above data.
9. Predict the value of \[\hat{Y}\] when X = 16
10. What is the maximum and minimum value for r?
11. If r ² is 1 what does it mean?
12. What is lower limit and upper limit of r ² ?
13. Complete table and sum each column

1. A large hotel purchased 200 new color televisions several months ago: 80 of one brand and 60 of each of two other brands. Records were kept for each set as to how many service calls were required, resulting in the table that follows.

Assume the TV sets are random samples of their brands. With 5% risk of Type I error, test for an association between TV brand and the number of service calls.

1. Is the \[{{\chi }^{2}}\] value significant at 5% level of significant?
2. Write the conclusion for this question

1. For n = 6 data points, the following quantities have been calculated:
    = 400 \(\sum{x}\) = 40 \(\sum{y}\)= 76 \(\sum{{{x}^{2}}}\) = 346 \(\sum{{{y}^{2}}}\)= 1160
   1. Determine the least squares regression line.
   2. Determine the standard error of estimate.
   3. Construct the 95% confidence interval for the mean of y when x = 7.0.
   4. Construct the 95% confidence interval for the mean of y when x = 9.0.
   5. Compared the width of the confidence interval obtained in part (c) with that obtained in part (d). Which is wider and why?

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