Q: Given $$\hat{p}_1 = 0.12$$, $$\hat{p}_2 = 0.04$$, $$n_1 = 1000$, n_2$ = 800$$, and $$z_{\alpha/2} = 1.96$$, what is the margin of error $$E$$ for a 95% confidence interval for $$p_1$$ - $$p_2$$?

$$E = 1.96 \sqrt{\dfrac{0.12 \times 0.88}{1000} + \dfrac{0.04 \times 0.96}{800}}$$

Q: Suppose you are testing $$H_0$$: $$p_1$$ = $$p_2$$ versus $$H_1$$: $$p_1$$ \neq $$p_2. If $$your calculated P-value is 0.03 and $$\alpha = 0.05$$, what is the correct conclusion?

Reject the null hypothesis; there is evidence that $$p_1$$ \neq $$p_2.$$

Question 1

Which of the following is the correct formula for the pooled sample proportion $$\bar{p}$$ when comparing two population proportions?

Accepted Answer

$$\bar{p} = \dfrac{x_1$ + x_2$}{n_1$ + n_2$}$$

Question 2

A researcher wants to test if the proportion of smokers differs between two cities. Which requirement must be satisfied for the two-sample proportion z-test to be valid?

Accepted Answer

Each sample must have at least 5 successes and 5 failures.

Question 3

Given $$\hat{p}_1 = 0.12$$, $$\hat{p}_2 = 0.04$$, $$n_1 = 1000$, n_2$ = 800$$, and $$z_{\alpha/2} = 1.96$$, what is the margin of error $$E$$ for a 95% confidence interval for $$p_1$$ - $$p_2$$?

Accepted Answer

$$E = 1.96 \sqrt{\dfrac{0.12 	imes 0.88}{1000} + \dfrac{0.04 	imes 0.96}{800}}$$

Question 4

In the context of comparing two independent means, which of the following is the correct test statistic when population standard deviations are unknown and not assumed equal?

Accepted Answer

$$t = \dfrac{\bar{x}_1 - \bar{x}_2 - (\mu_1$ - \mu_2$)}{\sqrt{\dfrac{s_1$^2}{n_1$} + \dfrac{s_2$^2}{n_2$}}}$$

Question 5

A study compares the mean blood pressure before and after a new medication in the same group of patients. Which statistical method is most appropriate?

Accepted Answer

Paired (matched pairs) t-test for dependent samples

Question 6

Which of the following statements about the confidence interval for the difference between two proportions is correct?

Accepted Answer

If the confidence interval does not contain 0, there is evidence of a significant difference between the two proportions.

Question 7

Suppose you are testing $$H_0$$: $$p_1$$ = $$p_2$$ versus $$H_1$$: $$p_1$$ 
eq $$p_2. If $$your calculated P-value is 0.03 and $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is evidence that $$p_1$$ 
eq $$p_2.$$

Question 8

Which of the following is a requirement for using the two-sample t-test for independent means?

Accepted Answer

The two samples must be independent random samples.

Question 9

A researcher collects paired data on the weights of patients before and after a diet program. What is the correct formula for the test statistic to test if the mean difference is zero?

Accepted Answer

$$t = \dfrac{\bar{d} - 0}{s_d / \sqrt{n}}$$

Question 10

If the requirements for the two-sample proportion z-test are not met (e.g., fewer than 5 successes or failures in a sample), which alternative method is recommended?

Accepted Answer

Fisher’s exact test

Question 11

In a study, the 95% confidence interval for $$\mu_1$ - \mu_2$ is (-0.53$$, 0.27)$$. What does this interval suggest about the difference between the two population means?

Accepted Answer

There is not a significant difference between the two means.

Question 12

Which formula gives the variance of the difference between two independent sample proportions?

Accepted Answer

$$\$$sigma^2$$_{(\hat{p}_1 - \hat{p}_2)} = \dfrac{p_1$ q_1$}{n_1$} + \dfrac{p_2$ q_2$}{n_2$}$$

Question 13

Which of the following best describes a scenario where two samples are dependent?

Accepted Answer

Blood pressure measured before and after treatment in the same patients.

Question 14

When constructing a confidence interval for the difference between two independent means, which distribution is used if the population standard deviations are unknown?

Accepted Answer

Student’s t-distribution

Question 15

A study finds a P-value of 0.0000 when comparing two proportions. What does this indicate about the null hypothesis?

Accepted Answer

There is extremely strong evidence against the null hypothesis.

Question 16

Which of the following is NOT a requirement for using the paired t-test for matched pairs?

Accepted Answer

The two samples must be independent.

Question 17

If you are constructing a confidence interval for the difference between two proportions and the requirements are not met, which method can be used?

Accepted Answer

Bootstrap resampling

Question 18

Which of the following best explains why using matched pairs (dependent samples) is often preferred in experimental design?

Accepted Answer

It controls for variability between subjects, increasing statistical power.

Inferences from Two Samples: Proportions and Means

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