Q: A 95% confidence interval for a population proportion is calculated as $$\hat{p} \pm z_{0.025} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. If$$ $$\hat{p} = 0.6$$ and $$n = 100$$, what is the margin of error? (Use $$z_{0.025} = 1.96)$$

$$1.96 \times \sqrt{\frac{0.6 \times 0.4}{100}}$$

Question 1

A pharmaceutical company claims that their new drug is effective in 80% of patients. In a clinical trial, 10 patients are treated. What is the probability that exactly 8 patients will respond to the drug? Use the binomial probability formula: $$P(X = k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$.

Accepted Answer

$$P(X = 8) = \binom{10}{8} (0.8)^8$ (0.2)^2$

Question 2

A researcher wants to test if a new teaching method improves test scores. She randomly assigns 30 students to the new method and 30 to the traditional method. Which hypothesis test should she use to compare the mean scores of the two groups?

Accepted Answer

Two-sample t-test

Question 3

Which of the following is the correct formula for a 95% confidence interval for the mean when the population standard deviation is unknown?

Accepted Answer

$$\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$$

Question 4

A hospital records whether patients recover (Yes/No) and whether they received a new treatment (Yes/No). Which statistical tool is most appropriate to analyze the association between treatment and recovery?

Accepted Answer

Contingency table and chi-square test

Question 5

In a regression analysis, the coefficient of determination $$r^2 is 0.81. $$What does this indicate?

Accepted Answer

81% of the variation in the dependent variable is explained by the model

Question 6

A study measures blood pressure before and after a new diet in the same group of patients. Which hypothesis test is most appropriate?

Accepted Answer

Paired t-test

Question 7

Which of the following is the correct formula for the test statistic in a one-sample z-test for the mean?

Accepted Answer

$$z = \frac{\bar{x} - \mu_0$}{\sigma / \sqrt{n}}$$

Question 8

A regression output gives a p-value of 0.03 for the slope coefficient. At a significance level of 0.05, what is the correct conclusion?

Accepted Answer

There is significant evidence that the slope is not zero

Question 9

A 95% confidence interval for a population proportion is calculated as $$\hat{p} \pm z_{0.025} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. If$$ $$\hat{p} = 0.6$$ and $$n = 100$$, what is the margin of error? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$1.96 	imes \sqrt{\frac{0.6 	imes 0.4}{100}}$$

Question 10

A contingency table is used to analyze the relationship between two categorical variables. Which of the following statements is true?

Accepted Answer

It displays the frequency distribution of the variables

Question 11

A researcher wants to estimate the mean weight of apples in an orchard. She collects a random sample of 50 apples and calculates a 99% confidence interval. What will happen to the width of the confidence interval if she increases the sample size to 200?

Accepted Answer

The confidence interval will become narrower

Question 12

Which of the following is the correct formula for the regression line?

Accepted Answer

$$\hat{y} = b_0$ + b_1$ x$$

Question 13

A study tests whether the mean cholesterol level in a population is different from 200 mg/dL. The sample mean is 210 mg/dL, the sample standard deviation is 20 mg/dL, and the sample size is 25. What is the value of the t-statistic?

Accepted Answer

$$t = \frac{210 - 200}{20 / \sqrt{25}} = 2.5$$

Question 14

Which of the following best describes the purpose of ANOVA in regression analysis?

Accepted Answer

To test whether the regression model explains a significant amount of variation in the dependent variable

Question 15

A company wants to know if the proportion of defective products is less than 5%. Which hypothesis test should they use?

Accepted Answer

One-sample z-test for proportions

Question 16

If the p-value for a hypothesis test is 0.08 and the significance level is 0.05, what is the correct conclusion?

Accepted Answer

Fail to reject the null hypothesis

Question 17

Which of the following is NOT an assumption of the simple linear regression model?

Accepted Answer

The independent variable is normally distributed

Question 18

A binomial experiment consists of 12 trials with probability of success $$p = 0.3. $$What is the expected number of successes?

Accepted Answer

$$E[X] = np = 12 	imes 0.3 = 3.6$$

Question 19

A medical researcher is comparing the mean recovery times for two different drugs using independent samples. The sample sizes are both greater than 30. Which distribution should be used to calculate the p-value?

Accepted Answer

Standard normal (z) distribution

Question 20

Which of the following statements about confidence intervals is correct?

Accepted Answer

A 95% confidence interval means that if we repeated the sampling process many times, about 95% of the intervals would contain the true parameter

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