To test whether the sample comes from a population with a mean of 38, we will conduct a hypothesis test for the population mean using the following steps:

Step 1: State the hypotheses

• Null hypothesis $H_0$: The population mean is 38. ($\mu = 38$)
• Alternative hypothesis $H_1$: The population mean is not 38. ($\mu \neq 38$)

This is a two-tailed test because we are testing if the sample mean is different (either higher or lower) from 38.

Step 2: Choose the significance level

The significance level is typically 0.05 for a 95% confidence level.

Step 3: Calculate the test statistic

Since we know the population standard deviation, we will use the Z-test formula for the population mean:

$Z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$

Where:

• $\bar{x}$ = sample mean = 40
• $\mu_0$ = hypothesized population mean = 38
• $\sigma$ = population standard deviation = 10
• $n$ = sample size = 400

Substituting the values:

$Z = \frac{40 - 38}{\frac{10}{\sqrt{400}}} = \frac{2}{\frac{10}{20}} = \frac{2}{0.5} = 4$

Step 4: Find the critical value

For a two-tailed test at the 95% confidence level, the critical value of $Z$ at $\alpha = 0.05$ is approximately ±1.96 (from standard Z-tables).

Step 5: Make a decision

• If the absolute value of the test statistic $|Z|$ is greater than 1.96, we reject the null hypothesis.

In this case, $|Z| = 4$, which is greater than 1.96, so we reject the null hypothesis.

Conclusion for the hypothesis test:

The sample provides sufficient evidence to conclude that the population mean is not 38.

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Step 6: Calculate the 95% Confidence Interval

To calculate the 95% confidence interval for the population mean, we use the formula:

$\text{CI} = \bar{x} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$

Where:

• $\bar{x}$ = 40
• $Z_{\alpha/2}$ = 1.96 (critical value for 95% confidence)
• $\sigma$ = 10
• $n$ = 400

Substituting the values:

$\text{CI} = 40 \pm 1.96 \times \frac{10}{\sqrt{400}} = 40 \pm 1.96 \times 0.5 = 40 \pm 0.98$

Thus, the 95% confidence interval for the population mean is:

$(39.02, 40.98)$

Conclusion for the confidence interval:

We are 95% confident that the population mean lies between 39.02 and 40.98.

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Recap:

• Hypothesis test conclusion: The sample does not come from a population with mean 38.
• 95% confidence interval: The population mean is between 39.02 and 40.98.

Would you like more details on any part of this process? Here are some related questions you may find useful:

1. How do we interpret the results of a hypothesis test for the population mean?
2. What is the difference between a Z-test and a T-test for hypothesis testing?
3. How does the sample size affect the width of the confidence interval?
4. What assumptions are made when conducting hypothesis tests on population means?
5. How do we calculate the margin of error for a confidence interval?

Tip: In hypothesis testing, when sample size is large (like $n \geq 30$), we often use the Z-test even if the population standard deviation is unknown.