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

$=1.460$

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.