Question

A manufacturer of potato chips would like to know whether its bag filling machine works correctly at the 421 gram setting. It is believed that the machine is underfilling the bags. A 43 bag sample had a mean of 414 grams. Assume the population standard deviation is known to be 19.

Required:
a. Is there sufficient evidence at the 0.1 level that the bags are underfilled?
b. Find the P-value of the test statistic.

Answer 1

a) The p-value of the test is 0.0078 < 0.1, which means that there is sufficient evidence at the 0.1 level that the bags are underfilled.

b) 0.0078.

Explanation:

Question a:

A manufacturer of potato chips would like to know whether its bag filling machine works correctly at the 421 gram setting.

At the null hypothesis, it is tested if the mean is of 421, that is:

`H_0: \mu = 421`

It is believed that the machine is underfilling the bags.

At the alternative hypothesis, it is tested if the mean is of less than 421, that is:

`H_a: \mu < 421`

The test statistic is:

`z = (X - \mu)/((\sigma)/(√(n)))`

In which X is the sample mean,
`\mu` is the value tested at the null hypothesis,
`\sigma` is the standard deviation and n is the size of the sample.

421 is tested at the null hypothesis:

This means that
`\mu = 421`

A 43 bag sample had a mean of 414 grams. Assume the population standard deviation is known to be 19.

This means that
`n = 43, X = 414, \sigma = 19`

Value of the test statistic:

`z = (X - \mu)/((\sigma)/(√(n)))`

`z = (414 - 421)/((19)/(√(43)))`

`z = -2.42`

P-value of the test and decision:

The p-value of the test is the probability of finding a sample mean below 414, which is the p-value of z = -2.42.

Looking at the z-table, z = -2.42 has a p-value of 0.0078.

The p-value of the test is 0.0078 < 0.1, which means that there is sufficient evidence at the 0.1 level that the bags are underfilled.

b. Find the P-value of the test statistic.

As found above, the p-value of the test statistic is 0.0078.