Step 1: Identify the Hypotheses

The claim in this case is that the population mean ($\mu$) is equal to 51,500. This suggests we are testing whether the population mean is equal to 51,500 or not.

The null hypothesis ($H_0$) is that the population mean is equal to 51,500, and the alternative hypothesis ($H_a$) is that the population mean is not equal to 51,500. This represents a two-tailed test.

Thus, the correct hypotheses are:

• $H_0: \mu = 51,500$ (null hypothesis)
• $H_a: \mu \neq 51,500$ (alternative hypothesis)

The correct answer is: C. $H_0: \mu = 51,500$, $H_a: \mu \neq 51,500$.

Step 2: Calculate the Test Statistic

We use the t-test statistic formula for a sample mean:

$t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$

Where:

• $\bar{x} = 50,635$ (sample mean)
• $\mu_0 = 51,500$ (claimed population mean)
• $s = 2,800$ (sample standard deviation)
• $n = 19$ (sample size)

Substitute the values into the formula:

$t = \frac{50,635 - 51,500}{\frac{2,800}{\sqrt{19}}}$

Step 3: Compute the Denominator

$\frac{2,800}{\sqrt{19}} \approx \frac{2,800}{4.3589} \approx 642.35$

Step 4: Compute the Test Statistic

Now, calculate the numerator:

$50,635 - 51,500 = -865$

Now, calculate the entire expression:

$t = \frac{-865}{642.35} \approx -1.35$

Step 5: Conclusion

The value of the standardized test statistic is approximately -1.35.

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Would you like me to help you with the next steps, such as finding the critical value or determining whether to reject the null hypothesis based on the t-distribution table?

Additional Questions:

1. How do you interpret a two-tailed test result?
2. What is the meaning of the significance level ($\alpha$) in hypothesis testing?
3. How can you calculate the p-value from the t-statistic?
4. What is the difference between a one-tailed and two-tailed t-test?
5. How do you determine the degrees of freedom for a t-test?

Tip: Always ensure your sample size is large enough when performing t-tests to guarantee reliable results. For smaller sample sizes, ensure that the population is normally distributed.