This is a hypothesis testing problem where we need to determine if the claim that the average income is $24,500 is correct based on a sample. The hypothesis test will be carried out using a p-value approach at a significance level $\alpha = 0.05$.

Here are the steps:

Step 1: State the null and alternative hypotheses

We are testing if the average income is different from $24,500, so:

• Null hypothesis ($H_0$): $\mu = 24,500$ (The average income is $24,500)
• Alternative hypothesis ($H_a$): $\mu \neq 24,500$ (The average income is not $24,500)

This is a two-tailed test because the alternative hypothesis uses "$\neq$".

Step 2: Collect the sample information

From the problem, we have the following:

• Sample mean ($\bar{x}$) = 24,350
• Population mean ($\mu_0$) = 24,500 (claimed population mean)
• Population standard deviation ($\sigma$) = 810
• Sample size ($n$) = 100
• Significance level ($\alpha$) = 0.05

Step 3: Calculate the test statistic

Since the population standard deviation is known, we use the Z-test formula for the test statistic:

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

Substitute the known values:

$Z = \frac{24,350 - 24,500}{\frac{810}{\sqrt{100}}} = \frac{-150}{\frac{810}{10}} = \frac{-150}{81} \approx -1.85$

Step 4: Find the p-value

For a two-tailed test, we find the p-value corresponding to the absolute value of the Z-statistic.

Using the Z-value of $-1.85$, we can find the p-value from Z-tables or using statistical software. The p-value for $Z = -1.85$ is approximately 0.0322 for one tail. Since this is a two-tailed test, we multiply by 2:

$p\text{-value} \approx 2 \times 0.0322 = 0.0644$

Step 5: Compare the p-value with the significance level

• If the p-value $\leq \alpha$, we reject the null hypothesis.
• If the p-value $> \alpha$, we fail to reject the null hypothesis.

In this case:

$p\text{-value} = 0.0644 > \alpha = 0.05$

Step 6: Conclusion

Since the p-value is greater than the significance level, we fail to reject the null hypothesis. This means there is not enough evidence to conclude that the average income is different from $24,500.

Would you like further details or clarification on any part of this solution?

Here are some related questions you might consider:

1. What is the difference between a one-tailed and two-tailed test?
2. How is the p-value related to the confidence level of a hypothesis test?
3. What is the impact of the sample size on the hypothesis test?
4. How do you interpret the standard deviation in the context of this problem?
5. Why do we use the Z-test instead of the t-test here?

Tip: The p-value approach gives you the probability of obtaining test results at least as extreme as the observed data, under the assumption that the null hypothesis is true. Always ensure you're comparing it against the correct significance level for valid conclusions.