2-sided alternative hypothesis

larz · #1 $Old$ June 27th, 2009, 09:29 AM

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 $t_{\frac{0.025}{2}}$ .

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

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

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

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

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

$\bar{x}=\frac{75.5}{10}=7.55$
$s^{2}=0.01056$
s=0.10274

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

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

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

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

Thank you for helping!

Random Variable · #2 $Old$ June 27th, 2009, 10:07 AM

You would only need to know $t_{0.025/2}$ if you were calculating a 97.5% confidence interval for $\mu$ .

larz · #3 $Old$ June 27th, 2009, 03:22 PM

I didn't understand that either. My book seems to use that for the two sided hypotheses when finding the confidence interval. Is there a different formula I should use? or the same without dividing $\alpha$ by 2.

The critical region equation for hypotheses one mean and variance unknown written in the book for H1: $\mu$ does not equal $\mu_{0}$ is $|\bar{x}-\mu_{0}| \ge t_{\frac{\alpha}{2}}(n-1)\frac{s}{\sqrt{n}}$

Random Variable · #4 $Old$ June 27th, 2009, 04:36 PM

Quote:

Originally Posted by larz $View Post$

I didn't understand that either. My book seems to use that for the two sided hypotheses when finding the confidence interval. Is there a different formula I should use? or the same without dividing $\alpha$ by 2.

The critical region equation for hypotheses one mean and variance unknown written in the book for H1: $\mu$ does not equal $\mu_{0}$ is $|\bar{x}-\mu_{0}| \ge t_{\frac{\alpha}{2}}(n-1)\frac{s}{\sqrt{n}}$

$\alpha = 0.05$ in your problem