Question 1

Which of the following is the correct null hypothesis when testing whether two population proportions are equal?

Accepted Answer

$$H_0$$: $$p_1$$ = $$p_2$$

Question 2

Suppose you are comparing the proportion of defective products from two factories. Factory A has 20 defects out of 200 products, and Factory B has 30 defects out of 300 products. What is the pooled sample proportion $$\hat{p}$$ used in the hypothesis test?

Accepted Answer

$$\hat{p} = \frac{20 + 30}{200 + 300} = 0.10$$

Question 3

Which formula correctly represents the test statistic for comparing two proportions?

Accepted Answer

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

Question 4

A researcher wants to estimate the difference in proportions of smokers between two cities. Which formula gives the confidence interval estimate for $$p_1$$ - $$p_2$$?

Accepted Answer

$$(\hat{p}_1 - \hat{p}_2) \pm E$$

Question 5

In the context of comparing two proportions, what is the requirement for using the normal approximation for the sampling distribution?

Accepted Answer

$$n_1$$\hat{p} \geq 5$$ and n_2$\hat{p} \geq 5$$

Question 6

A study compares the proportion of students passing a statistics exam in two universities. University A: 60 out of 100 pass; University B: 80 out of 150 pass. What is the value of $$\hat{p}_1 - \hat{p}_2$$?

Accepted Answer

$$0.60 - 0.533 = 0.067$$

Question 7

If the confidence interval for $$p_1$$ - $$p_2 is$$ $$(-0.05, 0.10)$$, what can you conclude at the given confidence level?

Accepted Answer

There is not enough evidence to conclude a significant difference between the two proportions.

Question 8

Which of the following is NOT a requirement for comparing two proportions using the methods described?

Accepted Answer

The two samples must have the same sample size.

Question 9

A right-tailed hypothesis test is being conducted to determine if $$p_1$$ \u003e $$p_2. $$The calculated $$z-$$statistic is $$0.45$$ and the $$p-$$value is $$0.328. At$$ $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Fail to reject the null hypothesis.

Question 10

When the requirements for the normal approximation are not met (e.g., $$n_1$$\hat{p} \u003c 5$$ or n_2$\hat{p} \u003c 5$$), which method is recommended for constructing a confidence interval?

Accepted Answer

Bootstrapping method

Question 11

In a study, the confidence interval for the difference in proportions is calculated as $$(-0.015, 0.048). $$What does this interval suggest about the difference between the two population proportions?

Accepted Answer

There is no significant difference between the two proportions at the given confidence level.

Question 12

Which of the following best describes the purpose of the pooled sample proportion $$\hat{p} in $$hypothesis testing for two proportions?

Accepted Answer

It provides a combined estimate of the population proportion under the null hypothesis.

Question 13

A medical researcher is comparing the effectiveness of two drugs. Drug A: 45 successes out of 60; Drug B: 30 successes out of 50. What is the pooled proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{45 + 30}{60 + 50} = 0.6818$$

Question 14

Which of the following is a correct interpretation of a $$p-$$value in the context of a two-proportion hypothesis test?

Accepted Answer

The probability of observing a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true.

Question 15

If the confidence interval for $$p_1$$ - $$p_2$$ does NOT contain zero, what does this indicate?

Accepted Answer

There is a statistically significant difference between the two proportions.

Question 16

Which of the following is a valid reason for failing to reject the null hypothesis in a two-proportion test?

Accepted Answer

The $$p-$$value is greater than the significance level $$\alpha.$$

Question 17

A study finds a $$z-$$statistic of $$-1.2$$ for a two-tailed test comparing two proportions. If the critical value is $$\pm 1.96$$, what is the correct conclusion?

Accepted Answer

Fail to reject the null hypothesis.

Question 18

Which of the following statements about the use of confidence intervals in comparing two proportions is correct?

Accepted Answer

A confidence interval that contains zero suggests no significant difference between the proportions.

Question 19

If two samples are not independent, which of the following is true regarding the comparison of their proportions?

Accepted Answer

The standard two-proportion methods are not valid.

Question 20

Which of the following is a correct step when using a calculator to perform a two-proportion $$z-$$test?

Accepted Answer

Enter the number of successes and sample sizes for both samples.

Hypothesis Testing and Confidence Intervals for Two Proportions

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