Questions: Radioactive fallout from testing atomic bombs drifted across a region. There were 210 people in the region at the time and 39 of them eventually died of cancer. Cancer experts estimate that one would expect only about 33 cancer deaths in a group this size. Assume the sample is a typical group of people. a) Is the death rate observed in the group unusually high? b) Does this prove that exposure to radiation increases the risk of cancer? State the null and alternative hypotheses. Choose the correct answer below. A. H0: p=0.1571 HA: p<0.1571 B. H0: p=0.1571 HA: p>0.1571 C. H0: p=0.1571 HA: P ≠ 0.1571 D. The assumptions and conditions are not met, so the test cannot proceed.

Solution Steps

We set up our null and alternative hypotheses as follows:

• Null Hypothesis: \( H_0: p = 0.1571 \)
• Alternative Hypothesis: \( H_A: p > 0.1571 \)

The test statistic \( Z \) is calculated using the formula: \[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Substituting the values:

• Sample proportion \( \hat{p} = \frac{39}{210} \approx 0.1857 \)
• Hypothesized proportion \( p_0 = 0.1571 \)
• Sample size \( n = 210 \)

Calculating \( Z \): \[ Z = \frac{0.1857 - 0.1571}{\sqrt{\frac{0.1571(1 - 0.1571)}{210}}} \approx 1.1377 \]

The P-value associated with the test statistic \( Z = 1.1377 \) is found to be: \[ \text{P-value} = 0.1276 \]

For a significance level \( \alpha = 0.05 \) in a one-tailed test, the critical value is: \[ Z_{\text{critical}} = 1.6449 \] Thus, the critical region is defined as \( Z > 1.6449 \).

We compare the P-value with the significance level:

• Since \( 0.1276 > 0.05 \), we do not reject the null hypothesis.

The death rate observed in the group is not unusually high. Therefore, we conclude that this does not prove that exposure to radiation increases the risk of cancer.

Final Answer