To solve this hypothesis testing problem, we will follow these steps:

Step 1: State the hypotheses

• Null Hypothesis ($H_0$): $\mu = 74.2$
• Alternative Hypothesis ($H_a$): $\mu > 74.2$

Step 2: Calculate the sample mean ($\bar{x}$) and sample standard deviation ($s$)

The data given is: $72$, $68.6$, $79.8$, $83.4$, $82$.

We need to calculate:

1. The sample mean $\bar{x}$
2. The sample standard deviation $s$

$\bar{x} = \frac{\sum{x_i}}{n}$ $s = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{n-1}}$

Step 3: Calculate the test statistic

Since the population standard deviation is unknown and the sample size is small, we will use the $t$-test statistic, calculated as: $t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$ where $\mu_0$ is the hypothesized population mean (74.2).

Step 4: Calculate the P-value

The P-value is determined by finding the probability of observing a test statistic at least as extreme as the one computed, assuming the null hypothesis is true. This involves finding the area under the $t$-distribution to the right of the calculated $t$-statistic.

Step 5: Make a decision

Compare the P-value to the significance level $\alpha = 0.10$ to decide whether to reject or fail to reject the null hypothesis.

Now, let's compute these steps.### Results:

1. Sample Mean ($\bar{x}$): $77.160$
2. Sample Standard Deviation ($s$): $6.504$
3. Test Statistic ($t$): $1.018$ (accurate to 3 decimal places)
4. P-value: $0.183$ (accurate to 3 decimal places)

Conclusion:

• P-value comparison: The P-value $0.183$ is greater than the significance level $\alpha = 0.10$.

• Decision: Since the P-value is greater than $\alpha$, we fail to reject the null hypothesis.

• Final Conclusion: There is not sufficient sample evidence to support the claim that the population mean is greater than 74.2.

Would you like any more details or have any further questions?

Here are 8 related questions:

1. How would the test statistic change if the sample mean were lower?
2. What if the sample size was larger? How would that affect the P-value?
3. How does the significance level $\alpha$ impact the hypothesis test?
4. Can we perform a two-tailed test instead? How would the hypotheses change?
5. How does knowing the population standard deviation change the hypothesis testing process?
6. What if the data were not normally distributed? How would that impact the test?
7. How would you interpret a test statistic of 0?
8. What are Type I and Type II errors in hypothesis testing?

Tip: Always ensure that the conditions for using a specific statistical test (like the t-test) are met before applying it.