A manufacturer of banana chips would like to know whether its bag filling machine works correctly at the 411 gram setting. It is believed that the machine is overfilling the bags. A 10 bag sample had a mean of 417 grams with a variance of 169. Assume the population is normally distributed. Is there sufficient evidence at the 0.02 level that the bags are overfilled?

Answer:
Given:
The Hypothesized Mean $ (\mu) = 411 $
The Sample Mean $ (x\bar) = 417 $
The Sample Variance $ (s^2) = 169 $

$\therefore $ The sample Standard deviation $(s) = \sqrt(169) = 13 $
The Significance Level $ (\alpha) = 0.02 $

Solution:
The null and alternative hypothesis:
$ H_0: \mu = 411 $
$ H_1: \mu > 411 $

The test statistic $(t): $
$t = \frac{\bar{x} – \mu}{\frac{s}{\sqrt{n}}} $
$ = \frac{417 – 411}{\frac{13}{\sqrt{10}}} $

$ = 1.460 $

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

The p-value:
$\text{p-value} = \text{P}(t_9 > 1.460) $

$ = 0.0891 $

The conclusion:
The p-value is greater than the significance level. Therefore, we fail to reject the null hypothesis. There is not enough evidence to support the claim that bags are overfilled.

Final Answer:
The null and alternative hypothesis:
$ H_0: \mu = 411 $
$ H_1: \mu > 411 $

The test statistic $(t) = 1.460 $

The p-value $= 0.0891 $

The conclusion:
The p-value is greater than the significance level. Therefore, we fail to reject the null hypothesis. There is not enough evidence to support the claim that bags are overfilled.