A manufacturer of trail mix would like to know whether its bag filling machine works correctly at the 375 gram setting. It is believed that the machine is overfilling the bags. A sample of 10 bags had a mean of 382 grams with a variance of 144. Assume the population is normally distributed. Is there sufficient evidence at the 0.05 level that the bags are overfilled?

Answer:
Given:
The Hypothesized Mean $ (\mu) = 375 $
The Sample Mean $ (\bar{x}) = 382 $
The Sample Variance $ (s^2) = 144 $
The Sample Size $ (n) = 10 $

$\therefore$ The Sample Standard Deviation $ (s) = \sqrt{144} = 12 $
The Significance Level $ (\alpha) = 0.05 $

Solution:
The null and alternative hypothesis:
$ H_0: \mu = 375 $
$ H_1: \mu > 375 $

The test statistic $ (t): $
$t = \frac{\bar{x} – \mu}{\frac{s}{\sqrt{n}}} $
$ = \frac{382 – 375}{\frac{12}{\sqrt{10}}} $

$ = 1.845 $

The degree of freedom $ (df): $
$ df = n – 1 $
$ = 10 – 1 $
$ = 9 $

The p-value:
$ \text{p-value} = \text{P}(t_{9} > 1.845) $

$ = 0.0491 $

The conclusion:
The p-value is approximately equal to the significance level. Therefore, we are on the threshold of rejecting the null hypothesis, suggesting that there might be sufficient evidence to support the claim that the bags are overfilled. However, this is a borderline case, and the decision could depend on additional context or considerations.

Final Answer:
The null and alternative hypothesis:
$ H_0: \mu = 375 $
$ H_1: \mu > 375 $

The test statistic $ (t) = 1.845 $

The p-value $ = 0.0491 $

The conclusion:
The p-value is approximately equal to the significance level. Therefore, there might be sufficient evidence to support the claim that the bags are overfilled. However, the decision is borderline and should be interpreted with caution.