To address this hypothesis testing problem, we will follow the same process as before:

Step 1: State the Hypotheses

• Null Hypothesis ($H_0$): $\mu = 64.1$
• Alternative Hypothesis ($H_a$): $\mu < 64.1$

Step 2: Calculate the Sample Mean ($\bar{x}$) and Sample Standard Deviation ($s$)

The data given is: $71.3$, $63.3$, $44.2$, $48.5$.

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

The test statistic for the $t$-test is calculated using: $t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$ where $\mu_0$ is the hypothesized population mean (64.1).

Step 4: Calculate the P-value

For a left-tailed test, the P-value is the area under the $t$-distribution to the left of the calculated $t$-statistic.

Step 5: Make a Decision

Compare the P-value to the significance level $\alpha = 0.01$ to decide whether to reject or fail to reject the null hypothesis.

Let's compute these steps.### Results:

1. Sample Mean ($\bar{x}$): $56.825$
2. Sample Standard Deviation ($s$): $12.651$
3. Test Statistic ($t$): $-1.150$ (accurate to 3 decimal places)
4. P-value: $0.167$ (accurate to 3 decimal places)

Conclusion:

• P-value comparison: The P-value $0.167$ is greater than the significance level $\alpha = 0.01$.

• 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 less than 64.1.

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

Here are 8 related questions:

1. What would happen if we increased the sample size while keeping the same mean and standard deviation?
2. How does the choice of $\alpha$ (significance level) affect the outcome of the hypothesis test?
3. What are the implications of a negative test statistic in a hypothesis test?
4. What assumptions must be satisfied to use a $t$-test in hypothesis testing?
5. How would the P-value change if the alternative hypothesis was $\mu > 64.1$ instead?
6. Can you explain the difference between a one-tailed and a two-tailed test?
7. What if the data were heavily skewed? How would that affect the validity of the test?
8. What are potential consequences of failing to reject a null hypothesis that is actually false?

Tip: In hypothesis testing, a smaller P-value indicates stronger evidence against the null hypothesis, but always consider the context and the chosen significance level.