Hypothesis Testing for Population Proportions – Step-by-Step Guidance

Q9. Cursed Movie: Testing the Cancer Rate Among Cast and Crew

Background

Topic: Hypothesis Testing for Population Proportions

This question asks you to test whether the cancer rate among cast and crew members of the movie The Conqueror is greater than the general population rate of 40%. You are to use a significance level of 0.05.

Key Terms and Formulas

• Null hypothesis ($H_0$): The population proportion is 0.40 ($p = 0.40$).

• Alternative hypothesis ($H_1$): The population proportion is greater than 0.40 ($p > 0.40$).

• Sample proportion ($\hat{p}$): $\hat{p} = \frac{x}{n}$, where $x$ is the number of successes and $n$ is the sample size.

• Test statistic (z): $z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}$

• Significance level ($\alpha$): 0.05

Step-by-Step Guidance

1. State the hypotheses clearly: $H_0: p = 0.40$ $H_1: p > 0.40$

2. Calculate the sample proportion: $\hat{p} = \frac{91}{220}$

3. Check the normal approximation conditions: $np \geq 5$ and $nq \geq 5$, where $q = 1 - p$ $np = 220 \times 0.40$ $nq = 220 \times 0.60$ Both should be at least 5.

4. Compute the standard error for the proportion: $\sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.40 \times 0.60}{220}}$

5. Set up the test statistic formula: $z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}$

Try solving on your own before revealing the answer!

Final Answer:

After calculating the sample proportion and the z-score, compare the z-score to the critical value for a one-tailed test at $\alpha = 0.05$ (typically $z_{0.05} \approx 1.645$). If the z-score exceeds this value, you reject the null hypothesis and conclude that the cancer rate is significantly greater than 40%.

This suggests that the cast and crew may have experienced a higher cancer rate than expected, possibly supporting the claim of a "curse."