Answer
Confidence interval: $4147.2\lt y\lt4652.4$
We are 95% confident that the mean 28-day strength of all concrete whose 7-day strength is 2550 psi is between 4147.2 and 4652.4 psi.
Work Step by Step
From problem 15 from Section 14.1:
$s_e=271.042$
$∑(x_i-x ̅)^2=1318.773^2=1739162$
$x ̅=\frac{2300+3390+2430+2890+3330+2480+3380+2660+2620+3340}{10}=2882$
$n=10$, so:
$d.f.=n-2=8$
$level~of~confidence=(1-α).100$%
$95$% $=(1-α).100$%
$0.95=1-α$
$α=0.05$
$t_{\frac{α}{2}}=t_{0.025}=2.306$
(According to Table VI, for d.f. = 8 and area in right tail = 0.025)
$Lower~bound=ŷ -t_{\frac{α}{2}}.s_e\sqrt {\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=4399.8-2.306\times271.042\sqrt {\frac{1}{10}+\frac{(2550-2882)^2}{1739162}}=4147.2$
$Upper~bound=ŷ +t_{\frac{α}{2}}.s_e\sqrt {\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=4399.8+2.306\times271.042\sqrt {\frac{1}{10}+\frac{(2550-2882)^2}{1739162}}=4652.4$
