Let's work through the steps to answer this question.

Problem Breakdown:

• Given Data:
  • Sample size, $n = 217$
  • Number of subjects who developed nausea, $X = 52$
  • Claimed population proportion $p_0 = 0.20$
  • Significance level $\alpha = 0.05$
• Objective:
  • Test the claim that more than 20% of users develop nausea. This suggests a one-tailed hypothesis test.

Step 1: Identify the Null and Alternative Hypotheses

Since the claim is that more than 20% of users develop nausea, we set up the hypotheses as:

• Null Hypothesis (H₀): $p = 0.20$
• Alternative Hypothesis (H₁): $p > 0.20$

Based on this, the correct answer for the null and alternative hypotheses is:

• Option C: $H_0: p = 0.20$, $H_1: p > 0.20$

Step 2: Test Statistic Calculation

The test statistic for a proportion is calculated using the formula: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$ where:

• $\hat{p}$ is the sample proportion, calculated as $\hat{p} = \frac{X}{n} = \frac{52}{217} \approx 0.2396$,
• $p_0 = 0.20$,
• $n = 217$.

Substituting in the values:

1. Calculate $\hat{p}$: $\hat{p} = 0.2396$
2. Calculate the standard deviation: $\sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.20 \times 0.80}{217}} \approx 0.0270$
3. Calculate $z$-score: $z = \frac{0.2396 - 0.20}{0.0270} \approx 1.47$

Step 3: Calculate the P-value

Since this is a one-tailed test with $H_1: p > 0.20$, we look up the P-value for $z = 1.47$ using the standard normal distribution table. The P-value for $z = 1.47$ is approximately 0.0708.

Summary of Answers:

1. Null and Alternative Hypotheses: Option C.
2. Test Statistic: $z \approx 1.47$
3. P-value: Approximately 0.071 (rounded to three decimal places).

Would you like further details or have any other questions?

Related Questions:

1. How do we interpret the P-value in terms of the significance level?
2. What does the test statistic value signify in this context?
3. How would the conclusions change if the significance level were 0.10 instead of 0.05?
4. Can we calculate a confidence interval for the true proportion based on this sample?
5. How can we perform this test if the sample size is very small?

Tip:

In hypothesis testing, always start by clearly defining your null and alternative hypotheses. This helps ensure you're performing the correct test and interpreting results accurately.