Q: A nutritionist wants to estimate the difference in calcium intake proportions between males and females to within 3 percentage points with 95% confidence. If no prior estimates are available, what is the minimum sample size required for each group? (Use $$z_{0.025} = 1.96$$ and $$ME = 0.03)$$

$$n = \left(\frac{1.96}{0.03}\$$right)^2$$ \times 0.5 \times 0.5 = 2135$$

Question 1

Suppose you are comparing the proportion of patients experiencing a side effect in two independent groups. Which of the following is the correct formula for the standard deviation of the sampling distribution of the difference between two sample proportions, $$\hat{p}_1$$ and $$\hat{p}_2$$?

Accepted Answer

$$\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1$} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2$}}$$

Question 2

A clinical trial compares the proportion of headaches in two groups: 64 out of 374 in the experimental group and 68 out of 376 in the control group. What is the sample proportion of headaches in the experimental group?

Accepted Answer

$$\frac{64}{374} = 0.171$$

Question 3

Which of the following is NOT an assumption required for constructing a confidence interval for the difference between two proportions?

Accepted Answer

The two groups must be dependent on each other.

Question 4

Given the following data from a clinical trial: 547 out of 2103 in the experimental group and 368 out of 1671 in the control group reported headaches. What is the pooled proportion $$\hat{p}_{pooled}$$?

Accepted Answer

$$\frac{547+368}{2103+1671} = 0.242$$

Question 5

When conducting a two-proportion z-test, what is the typical null hypothesis?

Accepted Answer

$$H_0$$: $$p_1$$ - $$p_2 = 0$

Question 6

A 90% confidence interval for the difference in proportions is $$[-0.056, 0.036]. $$What is the correct interpretation?

Accepted Answer

There is no significant difference between the two population proportions at the 0.10 significance level.

Question 7

Which formula is used to calculate the test statistic $$z$$ for the two-proportion z-test?

Accepted Answer

$$z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}_{pooled}(1-\hat{p}_{pooled})\left(\frac{1}{n_1$} + \frac{1}{n_2$}\right)}}$$

Question 8

If you want to estimate the difference between two proportions to within a margin of error $$ME$$ with $$(1-\alpha)100\%$$ confidence, and you have no prior estimates, what value should you use for each sample proportion in the sample size formula?

Accepted Answer

$$0.5$$

Question 9

Which of the following scenarios would violate the assumptions for using two-sample proportion methods?

Accepted Answer

Comparing the proportion of left-handedness in twins (paired data).

Question 10

A nutritionist wants to estimate the difference in calcium intake proportions between males and females to within 3 percentage points with 95% confidence. If no prior estimates are available, what is the minimum sample size required for each group? (Use $$z_{0.025} = 1.96$$ and $$ME = 0.03)$$

Accepted Answer

$$n = \left(\frac{1.96}{0.03}\$$right)^2$$ 	imes 0.5 	imes 0.5 = 2135$$

Question 11

Which of the following best describes the meaning of a p-value in the context of a two-proportion z-test?

Accepted Answer

The probability of observing a difference as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true.

Question 12

If the confidence interval for $$(p_1$ - p_2$) is$$ $$[0.032, 0.050]$$, what can you conclude?

Accepted Answer

Population 1 has a significantly higher proportion than population 2.

Question 13

Why is it important to check the success/failure condition when comparing two proportions?

Accepted Answer

To ensure the sampling distribution of the difference in proportions is approximately Normal.

Question 14

In a two-proportion z-test, when should you use the pooled proportion in the standard error calculation?

Accepted Answer

When the null hypothesis assumes the two population proportions are equal.

Question 15

Which of the following is a correct interpretation if the confidence interval for $$(p_1$ - p_2$) is $$entirely negative?

Accepted Answer

Population 1 has a significantly lower proportion than population 2.

Comparing Two Proportions: Confidence Intervals and Hypothesis Testing

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