PLEASE PROVIDE ACCURATE ANSWERS ONLY: Based on this information, and assuming th

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PLEASE PROVIDE ACCURATE ANSWERS ONLY:

Based on this information, and assuming that the regression assumptions hold, answer the questions in the table below.

1. What is the 90% confidence interval for the mean son's height (in centimeters) when the father's height is 185 centimeters? (Carry your intermediate computations to at least four decimal places, and round your answer to at least one decimal place.)

Lower limit:___________________

Upper limit:___________________

2. Consider (but do not actually compute) the 90% prediction interval for an individual value for son's height when the father's height is 185 centimeters. How would this prediction interval compare to the confidence interval computed above (assuming that both intervals are computed from the same sample data)?

Choose one response to answer the question below:

The prediction interval would be positioned to the left of the confidence interval. OR

The prediction interval would have the same center as, but would be wider than, the confidence interval. OR

The prediction interval would have the same center as, but would be narrower than, the confidence interval. OR

The prediction interval would be positioned to the right of the confidence interval. OR

The prediction interval would be identical to the confidence interval.

3. For the father's height values in this sample, 157 centimeters is more extreme than 185 centimeters is, that is, 157 is farther from the sample mean father's height than 185 is. How would the 90% confidence interval for the mean son's height when the father's height is 157 centimeters compare to the 90% confidence interval for the mean son's height when the father's height is 185 centimeters?

Choose one response to answer the question below.:

The interval computed from a father's height of 157 would be wider but have the same center. OR

The interval computed from a father's height of 157 would be narrower and have a different center. OR

The interval computed from a father's height of 157 would be narrower but have the same center. OR

The interval computed from a father's height of 157 would be wider and have a different center. OR

The intervals would be identical.

PLEASE PROVIDE ACCURATE ANSWERS ONLY:

Sir Francis Galton, in the late s, studied the relationship between the size of parents and the size of their offspring, finding that it may be possible to predict offspring size from parent size. In this spirit, we have collected bivariate data for pairs of human fathers and their (oldest, adult) sons. We have computed the least-squares regression equation for the data to be , with denoting the variable for the height (in centimeters) of a father and denoting the variable for the height (in centimeters) of the son.

Archie is centimeters tall. We have used the regression equation above to predict the height of his son. Now we are interested in both a prediction interval for this height and a confidence interval for the mean height of sons of fathers who are centimeters tall. We have computed the following for the data:

mean square error (MSE) ; ,
where denote the fathers' heights in the sample, and denotes their mean.

Based on this information, and assuming that the regression assumptions hold, answer the questions in the table below.

(If necessary, consult a list of formulas.) 1800

Explanation / Answer

SE of Confidence interval for mean =sqrt(22.27*0.0789) =1.3256

Regression equation is y_hat=87.03+0.54 x where x is the variable for the height (in centimeters) of a father and Y denoting the variable for the height (in centimeters) of the son.

For the mean son's height (in centimeters) when the father's height is 185 centimeters we have y_hat=87.03+0.54*185 = 186.93

Degree of freedom of error =n-2=17-2=15

For 90% confidence interval critical value =t(0.10,15)= 1.7531 using excel function =TINV(0.1,15)

1-Lower limit=y_hat- t(0.10,15)*SE(Confidence interval)=186.93-1.7531*1.3256=184.6061

Upper limit=y_hat- t(0.10,15)*SE(Confidence interval)=186.93+1.7531*1.3256=189.2539

2- The standard error of prediction of individual observation is higher than the mean, so the prediction interval standard error is higher than confidence interval. Hence, prediction interval is wider than the confidence interval. So answer is

The prediction interval would have the same center as, but would be wider than, the confidence interval.

3-When father’s height is 185, the predicted value is higher than157 cm father height while standard error would be lower as 187 is close to mean as compared to 157. So, prediction intervals for both have different centers and for 185 the prediction interval is narrower than 157. So answer is

The interval computed from a father's height of 157 would be wider and have a different center.

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