A sample of 39 observations is selected from a normal population. The sample mean is 43, and the population standard deviation is 6. Conduct the following test of hypothesis using the 0.02 significance level.

Sure, let's conduct the hypothesis test step by step.

Step 1: State the Hypotheses

• Null Hypothesis ($H_0$): The population mean ($\mu$) is equal to a specified value. Let's assume the specified value is $\mu_0 = 40$.
• Alternative Hypothesis ($H_1$): The population mean ($\mu$) is not equal to the specified value ($\mu_0$).

So, we have: $H_0: \mu = 40$ $H_1: \mu \neq 40$

Step 2: Set the Significance Level

The significance level ($\alpha$) is given as 0.02.

Step 3: Calculate the Test Statistic

Since the sample size is 39, which is greater than 30, we can use the Z-test for the hypothesis test. The test statistic for the Z-test is calculated using the formula: $Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}$

Where:

• $\bar{X}$ is the sample mean (43)
• $\mu_0$ is the population mean under the null hypothesis (40)
• $\sigma$ is the population standard deviation (6)
• $n$ is the sample size (39)

Plugging in the values: $Z = \frac{43 - 40}{6 / \sqrt{39}}$ $Z = \frac{3}{6 / \sqrt{39}}$ $Z = \frac{3}{6 / 6.244}$ $Z = \frac{3}{0.96}$ $Z \approx 3.125$

Step 4: Determine the Critical Value(s)

For a two-tailed test at the 0.02 significance level, we need to find the critical values that correspond to $\alpha/2 = 0.01$ in each tail of the standard normal distribution.

Using the Z-table, the critical values for $\alpha/2 = 0.01$ are approximately $\pm 2.33$.

Step 5: Make the Decision

• If the test statistic falls within the range of the critical values ($-2.33 < Z < 2.33$), we fail to reject the null hypothesis.
• If the test statistic falls outside this range ($Z < -2.33$ or $Z > 2.33$), we reject the null hypothesis.

In this case, the calculated test statistic $Z \approx 3.125$ is greater than 2.33.

Step 6: Conclusion

Since the test statistic $Z \approx 3.125$ falls outside the range of the critical values, we reject the null hypothesis at the 0.02 significance level. This suggests that there is sufficient evidence to conclude that the population mean is not equal to 40.