Q: A regression analysis yields the equation $$ \hat{y} = 2 + 3x . $$What is the predicted value of $$ y $$ when $$ x = 4 $$?

$$ \hat{y} = 2 + 3 \times 4 = 14 $$

Question 1

A pharmaceutical company wants to compare the effectiveness of two different drugs in lowering blood pressure. They randomly assign 50 patients to Drug A and 50 patients to Drug B. After 8 weeks, they record the reduction in blood pressure for each patient. Which statistical test is most appropriate to compare the mean reduction in blood pressure between the two groups?

Accepted Answer

A two-sample t-test for independent means

Question 2

Which of the following is the correct formula for the test statistic when testing the difference between two independent means, assuming equal variances?

Accepted Answer

$$ t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{s_p^2$ \left( \frac{1}{n_1$} + \frac{1}{n_2$} \right)}} $$

Question 3

A researcher wants to estimate the difference in proportions of smokers between two cities. She samples 200 people from each city and finds 40 smokers in City A and 60 in City B. Which formula should she use to calculate a 95% confidence interval for the difference in proportions?

Accepted Answer

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

Question 4

In a study measuring the effect of a new teaching method, students' test scores are recorded before and after the intervention. Which type of sampling is this, and what is the appropriate hypothesis test?

Accepted Answer

Dependent (paired) sampling; paired t-test

Question 5

Given the following paired data for before and after a diet: Before: [200, 220, 210], After: [190, 210, 200]. What is the mean of the differences (Before - After)?

Accepted Answer

$$ \frac{(200-190) + (220-210) + (210-200)}{3} = \frac{10 + 10 + 10}{3} = 10 $$

Question 6

A hospital claims that its new procedure reduces average recovery time. To test this, you collect recovery times from 30 patients before and after the procedure. Which is the correct null hypothesis for a paired t-test?

Accepted Answer

$$ H_0$: \mu_d = 0 $$

Question 7

Which of the following best describes the coefficient of determination ($$R^2) in a $$linear regression analysis?

Accepted Answer

$$ R^2$  is $$the proportion of the variance in the response variable explained by the explanatory variable.

Question 8

A regression analysis yields the equation $$ \hat{y} = 2 + 3x . $$What is the predicted value of $$ y $$ when $$ x = 4 $$?

Accepted Answer

$$ \hat{y} = 2 + 3 	imes 4 = 14 $$

Question 9

In the context of a least squares regression line, what does the slope represent?

Accepted Answer

The expected change in the response variable for a one-unit increase in the explanatory variable.

Question 10

A study finds $$ R^2$ = 0.81 $$ for the relationship between hours studied and exam score. How should this be interpreted?

Accepted Answer

81% of the variation in exam scores is explained by hours studied.

Question 11

Which R function would you use to test the difference between two independent means?

Accepted Answer

t.test

Question 12

A researcher uses RStudio to compare the proportion of left-handed students in two different universities. Which function should they use?

Accepted Answer

prop.test

Question 13

Given the regression equation $$ \hat{y} = 5 - 0.7x $$, what does the intercept represent in context?

Accepted Answer

The predicted value of $$ y $$ when $$ x = 0 .$$

Question 14

A nutritionist wants to know if a new diet plan changes cholesterol levels. She measures cholesterol before and after the diet for 25 participants. What is the formula for the test statistic for the mean of the differences?

Accepted Answer

$$ t = \frac{\bar{d} - \mu_0$}{s_d / \sqrt{n}} $$

Question 15

Which of the following is the correct formula for a residual in linear regression?

Accepted Answer

$$ e_i = y_i - \hat{y}_i $$

Question 16

A study compares the mean test scores of two independent groups. The sample means are 85 and 80, with standard deviations 5 and 6, and sample sizes 30 each. What is the standard error for the difference in means?

Accepted Answer

$$ \sqrt{\frac{5^2$}{30} + \frac{6^2$}{30}} = \sqrt{\frac{25}{30} + \frac{36}{30}} = \sqrt{2.033} \approx 1.427 $$

Question 17

Which of the following best describes a situation where you would use a test for the difference between two independent proportions?

Accepted Answer

Comparing the proportion of patients who recover with Drug A versus Drug B, where each patient receives only one drug.

Question 18

A researcher finds a p-value of 0.03 when testing the difference between two means. At a significance level of 0.05, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is evidence of a difference between the means.

Question 19

Which of the following statements about statistical inference is most accurate?

Accepted Answer

Statistical inference allows us to draw conclusions about populations based on sample data.

Statistical Inference: Comparing Two Populations and Linear Regression

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