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, then measure the reduction in blood pressure after 4 weeks. Which type of sampling is being used in this study?

Accepted Answer

Independent sampling

Question 2

Which of the following is the correct null hypothesis for testing whether there is a difference in the mean cholesterol levels before and after a new diet in the same group of patients?

Accepted Answer

$$H_0$$: \mu_{d} = 0$$

Question 3

A researcher is interested in the difference between the proportions of smokers in two different cities. Which formula should be used to calculate the standard error for the difference between two independent proportions?

Accepted Answer

$$SE = \sqrt{\frac{p_1$(1-p_1$)}{n_1$} + \frac{p_2$(1-p_2$)}{n_2$}}$$

Question 4

Given the following paired data for before and after a training program: Before: [80, 85, 90], After: [82, 88, 91]. What is the mean of the differences (After - Before)?

Accepted Answer

2

Question 5

A 95% confidence interval for the difference between two independent means is calculated as $$(-2.1, 5.3). $$What is the correct interpretation of this interval?

Accepted Answer

We are 95% confident that the true difference in means is between -2.1 and 5.3.

Question 6

Which R function would you use to test whether the mean difference in paired data is significantly different from zero?

Accepted Answer

t.test

Question 7

A study finds that the coefficient of determination ($$R^2) $$for the relationship between hours studied and exam score is 0.64. What does this mean?

Accepted Answer

64% of the variation in exam scores is explained by hours studied.

Question 8

Given the regression equation $$\hat{y} = 3 + 2x$$, what is the predicted value of $$y$$ when $$x = 5$$?

Accepted Answer

13

Question 9

In a linear regression analysis, 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 hospital wants to compare the recovery rates of two treatments for a disease. Out of 100 patients, 60 recover with Treatment A and 50 recover with Treatment B. What is the test statistic for the difference in proportions if $$p_1 = 0.6$, p_2$ = 0.5$$, $$n_1$$ = $$n_2 = 100$? Use the formula z$$ = \frac{$$p_1$$ - $$p_2$$}{\sqrt{\frac{$$p_1(1$$-$$p_1$$)}{$$n_1$$} + \frac{$$p_2(1$$-$$p_2$$)}{$$n_2$$}}}$$.

Accepted Answer

1.43

Question 11

Which of the following best describes a residual in linear regression?

Accepted Answer

The difference between the observed value and the predicted value.

Question 12

A researcher uses RStudio to analyze the difference in means between two groups and obtains a p-value of 0.03. What is the correct conclusion at the 0.05 significance level?

Accepted Answer

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

Question 13

Which of the following is the correct formula for a 95% confidence interval for the difference between two independent means?

Accepted Answer

$$(\bar{x}_1 - \bar{x}_2) \pm z^* \sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}$$

Question 14

A nutritionist wants to test if a new supplement changes the average weight of participants. She records the weights before and after taking the supplement for each participant. Which statistical test is most appropriate?

Accepted Answer

Paired t-test

Question 15

If the intercept of a least squares regression line is 5, what does this mean in context?

Accepted Answer

When the explanatory variable is zero, the predicted value of the response variable is 5.

Question 16

Which of the following statements about the unified nature of statistical inference is correct?

Accepted Answer

Both estimation and hypothesis testing use sample data to make inferences about a population.

Question 17

A study compares the mean test scores of students from two different schools. The sample means are 78 and 82, with standard deviations of 10 and 12, and sample sizes of 30 each. What is the standard error for the difference in means?

Accepted Answer

$$SE = \sqrt{\frac{10^2$}{30} + \frac{12^2$}{30}} = \sqrt{\frac{100}{30} + \frac{144}{30}} = \sqrt{3.33 + 4.8} = \sqrt{8.13} \approx 2.85$$

Question 18

Which of the following is NOT a correct interpretation of a p-value?

Accepted Answer

The probability that the null hypothesis is true.

Question 19

A scientist uses RStudio's prop.test function to compare the proportion of defective products from two factories. Which output from prop.test is most directly used to determine if there is a significant difference between the factories?

Accepted Answer

The p-value

Question 20

A regression analysis yields the equation $$\hat{y} = 4 + 1.5x. If a $$data point has $$x = 6$$ and an observed $$y$$ value of 15, what is the residual for this data point?

Accepted Answer

2

Statistical Inference: Comparing Two Populations and Linear Regression

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