Questions: State H₀ and Hₐ in words and in symbols. Then determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. Explain your reasoning. Sketch a normal sampling distribution and shade the area for the P-value. A report claims that lung cancer accounts for at most 39% of all cancer diagnoses. State the null hypothesis in words and in symbols. The null hypothesis expressed in words is the proportion of cancer diagnoses attributable to lung cancer is at most 39%. The null hypothesis is expressed symbolically as H₀: P ≤ 0.39.

Solution Steps

The null hypothesis (\(H_0\)) and alternative hypothesis (\(H_a\)) are defined as follows:

• Null Hypothesis: \(H_0: p \leq 0.39\) (The proportion of cancer diagnoses attributable to lung cancer is at most 39%)
• Alternative Hypothesis: \(H_a: p > 0.39\) (The proportion of cancer diagnoses attributable to lung cancer is more than 39%)

Since the alternative hypothesis tests whether the proportion is greater than 39%, this is a right-tailed test.

The test statistic is calculated using the formula:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Substituting the values:

• \(\hat{p} = 0.42\) (sample proportion)
• \(p_0 = 0.39\) (hypothesized population proportion)
• \(n = 100\) (sample size)

The calculated test statistic is:

\[ Z = 0.6151 \]

The P-value associated with the test statistic is:

\[ \text{P-value} = 0.2693 \]

For a significance level of \(\alpha = 0.05\) in a right-tailed test, the critical value is:

\[ Z > 1.6449 \]

To make a decision, we compare the test statistic to the critical value:

• Test Statistic: \(Z = 0.6151\)
• Critical Value: \(Z > 1.6449\)

Since \(0.6151 < 1.6449\), we fail to reject the null hypothesis. Additionally, the P-value \(0.2693\) is greater than \(\alpha = 0.05\).

Final Answer