Questions: Suppose 209 subjects are treated with a drug that is used to treat pain and 55 of them developed nausea. Use a 0.05 significance level to test the claim that more than 20% of users develop nausea. Identify the null and alternative hypotheses for this test. Choose the correct answer below. A. H0: p=0.20 H1: p>0.20 B. H0: p>0.20 H1: p=0.20 C. H0: p=0.20 H1: p<0.20 D. H0: p=0.20 H1: p ≠ 0.20

Suppose 209 subjects are treated with a drug that is used to treat pain and 55 of them developed nausea. Use a 0.05 significance level to test the claim that more than 20% of users develop nausea.

Identify the null and alternative hypotheses for this test. Choose the correct answer below.
A. H0: p=0.20 H1: p>0.20
B. H0: p>0.20 H1: p=0.20
C. H0: p=0.20 H1: p<0.20
D. H0: p=0.20 H1: p ≠ 0.20

Transcript text: Suppose 209 subjects are treated with a drug that is used to treat pain and 55 of them developed nausea. Use a 0.05 significance level to test the claim that more than $20 \%$ of users develop nausea. Identify the null and alternative hypotheses for this test. Choose the correct answer below. A. $\mathrm{H}_{0}: p=0.20$ $H_{1}: p>0.20$ B. $H_{0}: p>0.20$ $H_{1}: p=0.20$ C. $H_{0}: p=0.20$ $H_{1}: p<0.20$ D. $H_{0}: p=0.20$ $H_{1}: p \neq 0.20$

Solution

Solution Steps

To test the claim that more than 20% of users develop nausea, we need to set up the null and alternative hypotheses. The null hypothesis (H0) represents the status quo or the claim to be tested, while the alternative hypothesis (H1) represents the claim we are trying to find evidence for. In this case, the null hypothesis would be that the proportion of users developing nausea is 20%, and the alternative hypothesis would be that the proportion is greater than 20%.

Solution Approach

Define the null hypothesis (H0) and the alternative hypothesis (H1).
Use the sample data to calculate the test statistic.
Determine the p-value based on the test statistic.
Compare the p-value to the significance level (0.05) to decide whether to reject the null hypothesis.

Step 1: Define the Hypotheses

We need to test the claim that more than $20\%$ of users develop nausea. Therefore, we set up the null and alternative hypotheses as follows:

Null hypothesis: $ H_0: p = 0.20 $
Alternative hypothesis: $ H_1: p > 0.20 $

Step 2: Calculate the Sample Proportion

The sample proportion $ \hat{p} $ is calculated as: \[ \hat{p} = \frac{x}{n} = \frac{55}{209} \approx 0.2632 \]

Step 3: Calculate the Standard Error

The standard error (SE) is calculated using the formula: \[ \text{SE} = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.20 \times (1 - 0.20)}{209}} \approx 0.02767 \]

Step 4: Calculate the Test Statistic

The test statistic (z-score) is calculated as: \[ z = \frac{\hat{p} - p_0}{\text{SE}} = \frac{0.2632 - 0.20}{0.02767} \approx 2.2827 \]

Step 5: Determine the p-value

The p-value for a one-tailed test is calculated as: \[ \text{p-value} = 1 - \Phi(z) \approx 1 - \Phi(2.2827) \approx 0.01123 \]

Step 6: Compare the p-value to the Significance Level

We compare the p-value to the significance level $ \alpha = 0.05 $: \[ \text{p-value} = 0.01123 < 0.05 \]

Since the p-value is less than the significance level, we reject the null hypothesis.