You provided a set of hypothesis testing problems, both in Part B (listing multiple hypothesis tests with details like sample sizes and proportions) and in Part C (where two specific real-world scenarios need to be tested).

Let's break down how to approach these problems.

Part B: Hypothesis Testing Overview

For each of the hypothesis tests in Part B (4–10):

1. Null Hypothesis (H₀): The status quo or default assumption (like no difference or no effect).
2. Alternative Hypothesis (Hₐ): The claim or hypothesis you want to test (often claiming a difference or effect).
3. Significance Level (α): The probability of rejecting the null hypothesis when it's actually true (commonly set at 0.05, or 5%).

Example Walkthrough: Problem 4

• Hypothesis: H₀: P = 0.5 versus Hₐ: P < 0.5
• α = 0.05, sample size $n = 1200$, and $x = 480$.

Steps:

1. Calculate sample proportion: $\hat{P} = \frac{x}{n} = \frac{480}{1200} = 0.4$.
2. Use the Z-test for proportions to determine if the sample proportion is significantly less than 0.5.
3. Use the standard error formula for proportions:
   $SE = \sqrt{\frac{P(1 - P)}{n}} = \sqrt{\frac{0.5(1 - 0.5)}{1200}} = 0.0144$
4. Calculate the Z-score:
   $Z = \frac{\hat{P} - P}{SE} = \frac{0.4 - 0.5}{0.0144} = -6.94$
5. Compare this Z-score to the critical value at $\alpha = 0.05$ (one-tailed Z-test).

I can apply similar steps to the other problems if you'd like detailed solutions for each.

Part C: Real-World Hypothesis Testing

1. Factory Light Bulb Lifespan

• Claim: The average lifespan of LED bulbs is 50,000 hours.
• Sample: $n = 50$, mean lifespan $\bar{x} = 40,000$, standard deviation $s = 1,000$.
• Hypotheses:
  • $H_0$: $\mu = 50,000$ (the manufacturer's claim is correct).
  • $H_a$: $\mu \neq 50,000$ (the actual mean differs).

Perform a t-test:

1. Calculate the t-statistic: $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} = \frac{40,000 - 50,000}{\frac{1,000}{\sqrt{50}}} = -70.71$
2. Compare this to the critical t-value for $\alpha = 0.05$, two-tailed, with $n-1 = 49$ degrees of freedom.

2. Graduates' Job Search Time

• Claim: It takes graduates more than 3 months to find a job.
• Sample: $n = 36$, sample mean $\bar{x} = 3.5$, sample standard deviation $s = 1.5$.
• Hypotheses:
  • $H_0$: $\mu = 3$ (the average is 3 months or less).
  • $H_a$: $\mu > 3$ (it takes longer than 3 months on average).

Perform a t-test:

1. Calculate the t-statistic: $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} = \frac{3.5 - 3}{\frac{1.5}{\sqrt{36}}} = 2$
2. Compare the t-statistic to the critical value for a one-tailed test at $\alpha = 0.10$, with $df = 35$.

Would you like detailed solutions for each problem, or further clarification on any specific step? Here are 5 related questions to think about:

1. How do we decide between a one-tailed and two-tailed test in hypothesis testing?
2. What factors affect the power of a hypothesis test?
3. What is the relationship between sample size and the standard error?
4. How can hypothesis testing be used to evaluate claims in everyday situations?
5. What are the assumptions of using a t-test versus a z-test?

Tip: Always make sure your data meets the assumptions of the test you're using (e.g., normality for t-tests).