Questions: A data set includes weights (in grams) of 36 Reese's Peanut Butter Cup Miniatures. The accompanying Statdisk display shows results from using all 36 weights to test the claim that the sample is from a population with a mean equal to 8.953 g. Test the given claim by using the display provided from Statdisk. Use a 0.01 significance level. Identify the null and alternative hypotheses.

Solution Steps

We are testing the claim that the sample is from a population with a mean equal to $\mu_0 = 8.953 \, g$. The null and alternative hypotheses are defined as follows:

$$\mathrm{H}_{0}: \mu = 8.953$$ $$\mathrm{H}_{1}: \mu \neq 8.953$$

The standard error $SE$ is calculated using the formula:

$$SE = \frac{\sigma}{\sqrt{n}} = \frac{0.132}{\sqrt{36}} = 0.022$$

The test statistic $Z_{\text{test}}$ is calculated using the formula:

$$Z_{\text{test}} = \frac{\bar{x} - \mu_0}{SE} = \frac{8.941 - 8.953}{0.022} = -0.5455$$

For a two-tailed test, the p-value $P$ is calculated as:

$$P = 2 \times (1 - T(|z|)) = 0.5854$$

We compare the p-value to the significance level $\alpha = 0.01$:

• If $P < \alpha$, we reject the null hypothesis.
• If $P \geq \alpha$, we fail to reject the null hypothesis.

In this case:

$$0.5854 \geq 0.01$$

Since the p-value is greater than the significance level, we fail to reject the null hypothesis.

Final Answer