To solve this problem using the critical value approach, follow these steps:

Step 1: Set up the hypotheses and level of significance

• Null hypothesis ($H_0$): $\mu = 20$
• Alternative hypothesis ($H_a$): $\mu \neq 20$
• Level of significance ($\alpha$): 5% or 0.05

This is a two-tailed test, as the alternative hypothesis specifies $\mu \neq 20$.

Step 2: Calculate degrees of freedom and find critical values

The formula for degrees of freedom ($df$) is: $df = n - 1 = 15 - 1 = 14$

Using a $t$-distribution table or a calculator, for $df = 14$ and a two-tailed test with $\alpha = 0.05$: $t_{\text{critical}} = \pm 2.145$

Step 3: Compare the test statistic to the critical values

The test statistic provided is $t \approx 4.841$.

• The test statistic $t = 4.841$ is greater than the upper critical value $+2.145$.
• Therefore, the test statistic falls in the rejection region.

Step 4: Conclusion

Since the test statistic lies in the rejection region, we reject the null hypothesis ($H_0$).

Interpretation

At the 5% significance level, there is sufficient evidence to conclude that the population mean is not equal to 20.

Correct answer:

a) Reject the null hypothesis. At the 5% significance level, the data provide sufficient evidence that the population mean does not equal 20.

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Let me know if you'd like a more detailed explanation of the steps or any part of this process!

Related questions:

1. What would happen to the conclusion if the significance level were 1% instead of 5%?
2. How does the sample size affect the critical value in a $t$-test?
3. Why do we use a $t$-test instead of a $z$-test in this scenario?
4. How is the rejection region determined for a two-tailed test?
5. What is the impact of a larger standard deviation on the test statistic?

Tip:

Always check whether the assumptions for using the $t$-test are met, such as normality of the data or sufficiently large sample size.