To solve this problem, let's go through each step in detail:

1. Determine the Critical Value

Since the sample size $n = 21$ is small and the population standard deviation is unknown, we use a t-distribution. The significance level is $\alpha = 0.01$, and we have a one-tailed test because $H_1: \mu > 84.9$.

• Degrees of freedom (df): $df = n - 1 = 21 - 1 = 20$
• Critical value: We need the t-score for a one-tailed test with $\alpha = 0.01$ and $df = 20$.

Looking up or calculating this critical t-value, we find: $t_{0.01, 20} \approx 2.5278$

So the critical value for this test is: $\text{critical value} = 2.5278$

2. Calculate the Test Statistic

The test statistic for a one-sample t-test is calculated as follows: $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$ where:

• $\bar{x} = 90$ (sample mean),
• $\mu = 84.9$ (hypothesized population mean),
• $s = 15.5$ (sample standard deviation),
• $n = 21$ (sample size).

Substitute the values: $t = \frac{90 - 84.9}{\frac{15.5}{\sqrt{21}}}$

Now, let's calculate this step-by-step:

1. Calculate the denominator: $\frac{15.5}{\sqrt{21}} \approx 3.3829$

2. Calculate the numerator: $90 - 84.9 = 5.1$

3. Divide to find $t$: $t = \frac{5.1}{3.3829} \approx 1.5074$

So the test statistic for this sample is: $\text{test statistic} = 1.5074$

Final Answers

• Critical value = 2.5278
• Test statistic = 1.5074

Would you like more details on interpreting the results or about similar hypothesis testing concepts?

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Related Questions:

1. How would the critical value change if we were conducting a two-tailed test?
2. What if the sample size was larger, say $n = 50$; how would that affect the test?
3. How do you determine if a one-tailed or two-tailed test is appropriate?
4. How would this calculation differ if the population standard deviation were known?
5. What is the effect of a different significance level, such as $\alpha = 0.05$, on the critical value?

Tip: In hypothesis testing, always check if you need a t-distribution or a z-distribution, especially when dealing with small samples and unknown population standard deviations.