Question

A cereal box filling machine is designed to release an amount of 16 ounces of cereal into each box, and the machine’s manufacturer wants to know of any departure from this setting. The engineers at the factory randomly sample 150 boxes of cereal and find a sample mean of 15.75 ounces. If we know from previous research that the population is normally distributed with a standard deviation of 1.46 ounces, is there evidence that the mean amount of cereal in each box is different from 16 ounces at 0.05 significance? State the hypotheses, list and check the conditions, calculate the test statistic, find the p-value, and make a conclusion in a complete sentence related to the scenario.

Answer 1

Using the z-distribution, it is found that since the p-value is less than 0.05, there is evidence that the mean amount of cereal in each box is different from 16 ounces at 0.05 significance.

What are the hypothesis tested?

At the null hypothesis, it is tested if the mean is not different to 16 ounces, that is:

$H_0: \mu = 16$

At the alternative hypothesis, it is tested if the mean is different, hence:

$H_1: \mu \neq 16$

What is the test statistic?

The test statistic is:

$z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

In which:

• $\overline{x}$ is the sample mean.

• $\mu$ is the value tested at the null hypothesis.

• $\sigma$ is the standard deviation of the population.

• n is the sample size.

The parameters for this problem are:

$\overline{x} = 15.75, \mu = 16, \sigma = 1.46, n = 150$.

Hence:

$z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

$z = \frac{15.75 - 16}{\frac{1.46}{\sqrt{150}}}$

z = -2.1

What is the p-value and the conclusion?

Using a z-distribution calculator, for a two-tailed test, as we are testing if the mean is different of a value, with z = -2.1, the p-value is of 0.0357.

Since the p-value is less than 0.05, there is evidence that the mean amount of cereal in each box is different from 16 ounces at 0.05 significance.

More can be learned about the z-distribution at https://brainly.com/question/16313918

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