A manufacturer of popcorn would like to know whether its bag filling machine works correctly at the 350 gram setting. It is believed that the machine is overfilling the bags. A 15 bag sample had a mean of 360 grams with a variance of 121. Assume the population is normally distributed. Is there sufficient evidence at the 0.01 level that the bags are overfilled?

Answer:
Given:
The Hypothesized Mean $ (\mu) = 350 $
The Sample Mean $ (\bar{x}) = 360 $
The Sample Variance $ (s^2) = 121 $
The Sample Size $ (n) = 15 $

$\therefore$ The Sample Standard Deviation $ (s) = \sqrt{121} = 11 $
The Significance Level $ (\alpha) = 0.01 $

Solution:
The null and alternative hypothesis:
$ H_0: \mu = 350 $
$ H_1: \mu > 350 $

The test statistic $ (t): $
$t = \frac{\bar{x} – \mu}{\frac{s}{\sqrt{n}}} $
$ = \frac{360 – 350}{\frac{11}{\sqrt{15}}} $

$ = 3.521 $

The degree of freedom $ (df): $
$ df = n – 1 $
$ = 15 – 1 $
$ = 14 $

The p-value:
$ \text{p-value} = \text{P}(t_{14} > 3.521) $

$ = 0.0017 $

The conclusion:
The p-value is less than the significance level. Therefore, we reject the null hypothesis. There is sufficient evidence to support the claim that the bags are overfilled.

Final Answer:
The null and alternative hypothesis:
$ H_0: \mu = 350 $
$ H_1: \mu > 350 $

The test statistic $ (t) = 3.521 $

The p-value $ = 0.0017 $

The conclusion:
The p-value is less than the significance level. Therefore, we reject the null hypothesis. There is sufficient evidence to support the claim that the bags are overfilled.