This question can be solved with softwares like R: The variables are x=height of

Question

This question can be solved with softwares like R:

The variables are x=height of father and y = height of corresponding son. The unit is centimetre (cm). For input into R, the data vectors for heights of fathers and corresponding heights of their sons are
x=c(169.4, 178.8, 185.6, 169.8, 171.2, 182.1, 167.8, 178.1, 177.5, 170.8, 169.2, 167.6, 168.9, 173.5, 179, 181.4, 166.5, 169.9, 171.6, 170.1, 171.8, 163.3, 160.2, 164.1, 160.7, 166.1, 179, 170.3, 168.1, 172.6, 181, 170.7, 172.2, 181.5, 168.5, 165.7, 170.6, 183.2, 174.5, 175.5, 168.6, 167.9, 174.2, 194.3, 174.9, 174.9, 184.8, 173.9, 175.6, 169.5, 169.1, 176.7, 173.6, 164.1, 175.1, 165.1, 172.6, 159.9, 170, 176.8, 165.1, 183.2, 170.2, 168, 164.6, 170.2, 169.3, 174.3, 155.3, 161.8, 166.6, 174.8, 166.1, 172.5, 177.1, 174.3, 181.3, 162.3, 163.2, 165.8, 161.4, 173.3, 176.6, 179.7, 174.1, 170.6, 176.9, 180.1, 167.1, 183.4, 173.6, 177.6, 180.1, 171.9, 168.5, 159.5, 172.3, 171, 183.2, 182.7, 171.2, 168.1, 174.7, 165.2, 176.8, 166.9, 173, 170.3, 171.9, 182.4, 169.4, 177.8, 164.4, 173, 175.6, 177, 173.5, 167.7, 181.9, 194.2, 160.9, 180, 178.6, 164.7, 183.2, 173.1, 168.4, 178.7, 182, 171.2, 163.4, 184.9, 175, 184.1, 178.5, 180.1, 176.2, 165, 175.8, 175, 184.3, 172.9, 176.4, 179.2, 169, 158.7, 167.3, 159.2, 173.8, 175.5, 165.2, 191.3, 166.5, 170, 183.6, 174.8, 159, 163.6, 157.4, 164.2, 172.6, 167.8, 172.8, 177.3, 174.7, 187.4, 170.2, 187.1, 186.5, 173.9, 184.1, 179.1, 171.9, 165.4, 166, 172.5, 182.2, 176.2, 167.3, 184)
and
y=c(165.4, 178.1, 185, 169, 170.8, 190.5, 174.4, 173.2, 186.1, 177.8, 177.6, 170, 177.5, 180.5, 186.6, 181.6, 171.2, 176.6, 168.9, 184.5, 166.8, 172, 164.9, 165, 171.8, 176.3, 180, 168.3, 164.6, 169.5, 171, 168.6, 171.2, 172.3, 181.4, 169.3, 178.5, 177.3, 177.1, 181.5, 176.7, 167.9, 179.4, 192.1, 178.8, 174.3, 182.6, 167, 176, 183.4, 172.2, 187.1, 178.2, 171.8, 186.9, 167.6, 179.6, 172.1, 177.7, 179.3, 157, 175.7, 176.5, 181.4, 172.8, 185.2, 165.5, 179.1, 164.7, 168.7, 172.9, 191, 178.4, 172.1, 179.4, 170.1, 185.2, 166.3, 165.5, 169.3, 163.3, 169.3, 174.9, 179.7, 174.3, 176, 175.2, 185.4, 180, 178.7, 179.9, 162.4, 183.2, 169.5, 169.2, 178.4, 177.5, 173, 188.6, 185.4, 174.3, 170.6, 174.7, 176.3, 177.1, 172.7, 175.8, 174.2, 171.6, 176.8, 168.1, 176.1, 164.9, 169.6, 182.7, 159.2, 169.3, 171.9, 176.1, 182.2, 164.1, 172.2, 174, 176.5, 177.6, 173.6, 181.9, 183.5, 173.5, 172, 156.4, 182.9, 177.6, 179.6, 183.5, 169.7, 183, 169.6, 181.5, 180.8, 189.3, 176.1, 165.7, 166.9, 183.9, 157.3, 182.6, 174.5, 170.6, 173.5, 162, 198, 166.8, 178.5, 176.2, 177.2, 172.1, 171, 171.8, 160, 184.4, 172.4, 176.1, 178.2, 179.3, 182.1, 175.2, 169.9, 176.4, 178.8, 183.2, 169.1, 168.1, 173.9, 164.4, 172.1, 171.1, 181.4, 174.6, 184.3)
For the questions below, use 3 decimal places.

Part d) For comparison, let's not use regression to get a confidence interval for the subpopulation mean. Consider instead the subset of fathers whose height is within 2.5 cm of x 178, including the endpoints. Based on the corresponding sons of these fathers obtain an approximate 95% confidence interval of the form The lower endpoint of the 95% confidence interval is The upper endpoint of the 95% confidence interval is

Explanation / Answer

here values which are 2.5 cm away from 178 are 175.5 to 180.5

for which corresponding y is given below

from here mean Ybar =177.133

and std deviation Sy(x) =7.009

n=33

hence lower endpoint of 95% CI =177.133-2*7.009/(33)1/2 =174.693

and upper endpoint of 95% CI =177.133+2*7.009/(33)1/2 =179.573

x y 175.5 181.5 175.5 173.5 175.6 176 175.6 182.7 175.8 181.5 176.2 183 176.2 181.4 176.4 165.7 176.6 174.9 176.7 187.1 176.8 179.3 176.8 177.1 176.9 175.2 177 159.2 177.1 179.4 177.3 178.2 177.5 186.1 177.6 162.4 177.8 176.1 178.1 173.2 178.5 183.5 178.6 174 178.7 183.5 178.8 178.1 179 186.6 179 180 179.1 169.1 179.2 166.9 179.7 179.7 180 172.2 180.1 185.4 180.1 183.2 180.1 169.7 mean 177.694 177.133 std deviation 1.498 7.009

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