Let me analyze the problem from the uploaded image and provide a detailed explanation.

Problem Summary:

The task involves hypothesis testing about workers' job satisfaction. It tests whether the proportion of satisfied workers (based on new sample data) is higher than 49% (the established value from prior studies). The significance level is $\alpha = 0.05$.

Steps and Explanation:

Part a: Verify $n \hat{p}(1 - \hat{p}) \geq 10$

• $n = 100$ (sample size), and $\hat{p} = 0.49$ (hypothesized proportion).
• Calculation: $n \hat{p}(1 - \hat{p}) = 100 \times 0.49 \times (1 - 0.49) = 25.0$ Since $25.0 \geq 10$, the condition is satisfied.

Part b: Hypotheses in Symbolic Form

• Null hypothesis ($H_0$): The proportion of satisfied workers has not increased. $H_0: p = 0.49$.
• Alternative hypothesis ($H_a$): The proportion of satisfied workers has increased. $H_a: p > 0.49$.

Part c: Compute the Test Statistic ($z$)

• Sample statistic: 37 satisfied workers out of 100 ($\hat{p} = 0.37$).
• Formula for $z$: $z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}$ Substituting: $z = \frac{0.37 - 0.49}{\sqrt{\frac{0.49 \times 0.51}{100}}} = \frac{-0.12}{\sqrt{0.002499}} = \frac{-0.12}{0.04999} \approx -2.40$

Part d: Compute the $p$-Value

• The test is one-tailed (since $H_a: p > 0.49$).
• From the $z$-table: $p = P(Z < -2.40) \approx 0.0082$

Part e: Decision Based on $\alpha = 0.05$

• Compare $p$-value $(0.0082)$ with $\alpha (0.05)$: $p < \alpha$ Therefore, reject the null hypothesis.

Part f: Conclusion

• There is sufficient evidence to support the claim that the proportion of workers satisfied with their job has increased.

Questions for Further Analysis:

1. Why is it important to verify $n \hat{p}(1 - \hat{p}) \geq 10$ before conducting the test?
2. How does the significance level ($\alpha$) impact the decision-making process in hypothesis testing?
3. What are the potential implications if a Type I error is made in this case?
4. Can the conclusions of this test be generalized to the entire population? Why or why not?
5. How would the test change if we were conducting a two-tailed test instead of a one-tailed test?

Tip:

Always ensure the assumptions for hypothesis testing (e.g., sample size, independence, normality) are checked before proceeding with calculations to maintain the validity of your results.