Question

In a test of the hypothesis $H_0:\mu =70$versus$H_a:\mu >70$, a sample of n = 100 observations possessed mean$\overline{x}=49.4$and standard deviation s = 4.1. Find and interpret the p-value for this test.

Short answer

The p-value for the test is 0.9279. The p-value indicates the probability of getting the test statistic is more than -1.46 when the claim $\mu =50$is true.

Step-by-step solution

Given information

The hypothesis test is: $H_0:\mu =50$versus $H_a:\mu >50$.

A random sample of size 100 is selected.

The sample mean is $\overline{x}=49.4$.

The sample standard deviation is s=4.1.

Computing the value of the test statistic

The test statistic is:

$$\begin{gathered} z=\frac{\overline{x}-\mu }{\frac{s}{\sqrt{n}}} \\ =\frac{49.4-50}{\frac{4.1}{\sqrt{100}}} \\ =\frac{-0.6}{\frac{4.1}{10}} \\ =\frac{-0.6}{0.41} \\ =-1.46 \end{gathered}$$

The test statistic is z = -1.46.

 Step 3: Computing the p-value

The p-value for the right-tailed test is:

$$\begin{gathered} p=P\left(Z>-1.46\right) \\ =1-P\left(Z⩽-1.46\right) \\ =1-0.0721 \\ =0.9279 \end{gathered}$$

The z-table is used to obtain the probability of a z-score less than or equal to -1.46.

Therefore, the p-value is 0.9279.

Interpretation of the p-value

The p-value indicates the probability of getting the test statistic is more than -1.46 when the claim$\mu =50$ is true.