Part 1: Identify the Null and Alternative Hypotheses

We are testing whether more than 20% of the subjects develop nausea, which means this is a one-tailed test. The claim being tested corresponds to the alternative hypothesis $H_1$, where $p > 0.20$.

The correct hypotheses are:

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

The correct answer is: C. $H_0: p = 0.20$, $H_1: p > 0.20$

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Part 2: Calculate the Test Statistic

We are performing a one-proportion z-test. The formula for the z-test statistic is:

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$

Where:

• $\hat{p} = \frac{51}{227} \approx 0.2247$ (sample proportion)
• $p_0 = 0.20$ (hypothesized proportion)
• $n = 227$ (sample size)

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Step 1: Compute the Standard Error

$\text{SE} = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.20 \times (1 - 0.20)}{227}} = \sqrt{\frac{0.16}{227}} = \sqrt{0.000704} \approx 0.0265$

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Step 2: Compute the Test Statistic

$z = \frac{0.2247 - 0.20}{0.0265} = \frac{0.0247}{0.0265} \approx 0.93$

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Final Answer

The test statistic for this hypothesis test is:

$z \approx 0.93$

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Let me know if you have questions or need further clarifications!
Here are five related questions you may find helpful:

1. How do you determine the critical value for a one-tailed z-test at the 0.01 significance level?
2. How is the p-value found from a z-score in hypothesis testing?
3. What assumptions must be satisfied for a one-proportion z-test to be valid?
4. How would the test change if the sample size or proportion of nausea cases were different?
5. What is the difference between a one-tailed and two-tailed hypothesis test?

Tip: Always compare the calculated z-value to the critical z-value to determine whether to reject the null hypothesis.