Assuming the rest of my work is correct for this question, I am having trouble on part c. I don't know if I have the right formula and, if I do, I don't know how to calculate $\displaystyle t_{\frac{0.025}{2}}$.

Let X equal the length of wood blocks manufactured. Assume distribution of X is $\displaystyle N(\mu,\sigma^{2})$. The greatest lenth is 7.5 inches. We shall test the null hypothesis $\displaystyle H_{0}: \mu=7.5$against a 2-sided alternative hypothesis using 10 observations.

a) Define test statistic and critical region for an$\displaystyle \alpha=0.05$ significance level.

test statistic- $\displaystyle t=\frac{\bar{x}-7.5}{\frac{s}{\sqrt{10}}}$

critical region- $\displaystyle |t|=\frac{|\bar{x}-7.5|}{\frac{s}{\sqrt{10}}} \ge t_{\frac{\alpha}{2}}(10-1)=2.262$

Calculate the value of the test statistic and give your decision using the following data (n=10)

7.65

7.60

7.65

7.70

7.55

7.55

7.40

7.40

7.50

7.50

$\displaystyle \bar{x}=\frac{75.5}{10}=7.55$

$\displaystyle s^{2}=0.01056$

s=0.10274

t=1.539

1.539 is not greater than 2.262, therefore, it fails to reject $\displaystyle H_{0}: \mu=7.5$

c) Is $\displaystyle \mu=7.50$ contained in a 95% confidence interval for $\displaystyle \mu$?

$\displaystyle \bar{x}+/- t_{\frac{0.025}{2}}(n-1)(\frac{s}{\sqrt{n}})$

As of now, I am using my values of $\displaystyle \bar{x}=7.55$, s=0.10274 and n=10

Thank you for helping!