1. The accompanying table gives the dry weights \((Y)\) of 11 chick embryos ranging in age from 6 to 16 days \((X)\). Also given in the table are the values of the common logarithms of the weights \((Z)\).

1. Observe the following two scatter diagrams. Describe the relationships between age \((X)\) and dry weight \((Y)\), and between age and \(\log _{10}\) dry weight \((Z)\).
   
   
2. State the simple linear regression models for these two regressions: \(Y\) regressed on \(X\), and \(Z\) regressed on \(X\).
3. Determine the least-squares estimates of each of the regression lines in part (b).
4. Sketch each estimated line on the appropriate scatter diagram. Which of the two regression lines has the better fit? Based on your answers to parts (a)-(c), is it more appropriate to run a linear regression of \(Y\) on \(X\), or of \(Z\) on \(X\) ? Explain.
5. For the regression that you chose as being more appropriate in part (d), find \(95 \%\) confidence intervals for the true slope and intercept. Interpret each interval with regard to the null hypothesis that the true parameter is 0 .
6. For the regression that you chose as being more appropriate in part (d), find a \(95 \%\) confidence and prediction bands. Using your sketch, find and interpret an approximate \(95 \%\) confidence interval on the mean response for an 8-day-old chick.

3. For married couples with one or more offspring, a demographic study was conducted to determine the effect of the husband's annual income (at marriage) on the time (in months) between marriage and the birth of the first child. The following table gives the husband's annual income (INC) and the time between marriage and the birth of the first child (TIME) for a hypothetical sample of 20 couples.

1. On the scatter diagram above, sketch by eye a line that fits the data reasonably well. Comment on the relationship between TIME \((Y)\) and INC \((X)\).
2. Determine the least-squares estimates of the slope \(\left(\beta_{1}\right)\) and intercept \(\left(\beta_{0}\right)\) for the straight-line regression of TIME \((Y)\) on INC \((X)\).
3. Draw the estimated regression line on the accompanying scatter diagram. Comment on how well this line fits the data.
4. Are any of the assumptions for straight-line regression clearly not satisfied in this example?
5. Test the null hypothesis that the true slope is 0 . Interpret the results of this test.
6. Can you suggest a model that would describe the TIME-INC relationship better than a straight line does?

4. A sociologist assigned to a correctional institution was interested in studying the relationship between intelligence and delinquency. A delinquency index (ranging from 0 to 50 ) was formulated to account for both the severity and the frequency of crimes committed, while intelligence was measured by IQ. The following table displays the delinquency index (DI) and IQ of a sample of 18 convicted minors.

1. Given that \(\hat{\beta}_{1}=-0.249\) and \(\hat{\beta}_{0}=52.273\), draw the estimated regression line on the accompanying scatter diagram.
2. How do you account for the fact that \(\hat{Y}=52.273\) when IQ \(=0\), even though the delinquency index goes no higher than 50 ?
3. Find a \(95 \%\) confidence interval for the true slope \(\beta_{1}\), using the fact that \(S_{Y \mid X}=7.704\) and \(S_{X}=16.192\)
4. Interpret this confidence interval with regard to testing the null hypothesis of zero slope at the \(\alpha=.05\) level.
5. Notice that the convicted minor with \(\mathrm{IQ}=134\) and DI \(=39.6\) appears to be quite out of place in the data. Decide whether this outlier has any effect on your estimate of the IQ-DI relationship, by looking at the graph for the fitted line obtained when the outlier is omitted. (Note that \(\hat{\beta}_{0}=70.846\) and \(\hat{\beta}_{1}=-0.444\) when the outlier is removed.)
6. Test the null hypothesis of zero slope when the outlier is removed, given that \(S_{Y \mid X}=\) \(4.933, S_{X}=14.693\), and \(n=17\). (Use \(\alpha=.05\).)
7. For these data, would you conclude that the delinquency index decreases as IQ increases?

5. Following the last congressional election, a political scientist attempted to investigate the relationship between campaign expenditures on television advertisements and subsequent voter turnout. The following table presents the percentage of total campaign expenditures delegated to television advertisements (TVEXP) and the percentage of registered voter turnout (VOTE) for a hypothetical sample of 20 congressional districts.

1. Determine the least-squares line of VOTE on TVEXP, and draw the estimated line on the accompanying scatter diagram.
2. Are any of the assumptions for straight-line regression clearly not satisfied in this example?
3. Test the null hypothesis that the true slope is 0 ; be sure to interpret your result.
4. Test the hypothesis \(\mu_{Y \mid X_{0}}=45\) for \(X_{0}=\bar{X}=36.99\). Interpret your result.
5. Calculate the corresponding \(95 \%\) confidence interval for \(\mu_{Y \mid 36.99}\), and interpret your result.

1. Use the data set of Problem 1 in Chapter 5 to answer the following questions.

1. Determine the ANOVA tables for the following regressions: ( 1 ) dry weight ( \(Y)\) on age \((X)\), and (2) \(\log _{10}\) dry weight \((Z)\) on age \((X)\). The following results will be helpful in reducing computation time:
   \(S_{Y \mid X}^{2}=0.23218, \quad \mathrm{SSY}=8.16811 \quad S_{Z \mid X}^{2}=0.0007838, \quad \mathrm{SSZ}=4.2276830\)
2. Use the tables in part (a) to perform the $F$ test for the significance of each straight-line regression. Interpret your results.

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