Q: A pharmaceutical company claims that 60% of patients experience relief after taking a new drug. In a clinical trial of 100 patients, 70 report relief. What is the value of the test statistic for testing $$H_0$$: p = 0.6$$ versus H_a$$: p \neq 0.6$$?

$$z = \frac{0.7 - 0.6}{\sqrt{0.6 \times 0.4 / 100}}$$

Q: A researcher wants to estimate the proportion of defective products in a factory. In a random sample of 400 products, 32 are defective. What is the 95% confidence interval for the true proportion of defectives?

$$0.08 \pm 1.96 \times \sqrt{0.08 \times 0.92 / 400}$$

Q: If two events A and B are independent, which of the following is true?

$$P(A \cap B) = P(A) \times P(B)$$

Q: A sample of 36 students has a mean test score of 80 with a standard deviation of 12. What is the 99% confidence interval for the population mean?

$$80 \pm 2.576 \times \frac{12}{\sqrt{36}}$$

Q: A fair die is rolled. What is the probability of rolling a number less than 5 or an even number?

$$P(\text{less than 5}) + P(\text{even}) - P(\text{less than 5} \cap \text{even})$$

Q: A quality control manager wants to test if the defect rate has changed from 5%. In a sample of 200 items, 16 are defective. What is the $$p-$$value for the test $$H_0$$: p = 0.05$$ vs H_a$$: p \neq 0.05$$?

Calculate $$z = \frac{0.08 - 0.05}{\sqrt{0.05 \times 0.95 / 200}}$$ and use NormalCDF to find the $$p-$$value.

Question 1

A pharmaceutical company claims that 60% of patients experience relief after taking a new drug. In a clinical trial of 100 patients, 70 report relief. What is the value of the test statistic for testing $$H_0$$: p = 0.6$$ versus H_a$$: p 
eq 0.6$$?

Accepted Answer

$$z = \frac{0.7 - 0.6}{\sqrt{0.6 	imes 0.4 / 100}}$$

Question 2

Which of the following statements about the Central Limit Theorem (CLT) is correct?

Accepted Answer

The sampling distribution of the sample mean approaches normality as sample size increases, regardless of the population's distribution.

Question 3

A researcher wants to estimate the proportion of defective products in a factory. In a random sample of 400 products, 32 are defective. What is the 95% confidence interval for the true proportion of defectives?

Accepted Answer

$$0.08 \pm 1.96 	imes \sqrt{0.08 	imes 0.92 / 400}$$

Question 4

If two events A and B are independent, which of the following is true?

Accepted Answer

$$P(A \cap B) = P(A) 	imes P(B)$$

Question 5

A hospital administrator wants to know if a new treatment changes the recovery rate. The null hypothesis is $$H_0$$: p = 0.5$$. If the true recovery rate is 0.6 but the administrator fails to reject H_0$, what type of error has occurred?

Accepted Answer

Type II error ($$\beta)$$

Question 6

A sample of 36 students has a mean test score of 80 with a standard deviation of 12. What is the 99% confidence interval for the population mean?

Accepted Answer

$$80 \pm 2.576 	imes \frac{12}{\sqrt{36}}$$

Question 7

Which of the following best describes the effect of increasing the sample size on a confidence interval for a mean?

Accepted Answer

The confidence interval becomes narrower because the standard error decreases.

Question 8

A fair die is rolled. What is the probability of rolling a number less than 5 or an even number?

Accepted Answer

$$P(	ext{less than 5}) + P(	ext{even}) - P(	ext{less than 5} \cap 	ext{even})$$

Question 9

A study finds a statistically significant difference in blood pressure between two groups, but the difference is only 0.5 mmHg. What is the best interpretation?

Accepted Answer

The result is statistically significant but may not be practically significant.

Question 10

Which formula is used to calculate the standard error of the sample proportion?

Accepted Answer

$$SE = \sqrt{p(1-p)/n}$$

Question 11

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the $$z-$$score for $$x = 130$$?

Accepted Answer

$$z = \frac{130 - 100}{15}$$

Question 12

In a two-proportion hypothesis test, which formula is used for the pooled proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{x_1$ + x_2$}{n_1$ + n_2$}$$

Question 13

A quality control manager wants to test if the defect rate has changed from 5%. In a sample of 200 items, 16 are defective. What is the $$p-$$value for the test $$H_0$$: p = 0.05$$ vs H_a$$: p 
eq 0.05$$?

Accepted Answer

Calculate $$z = \frac{0.08 - 0.05}{\sqrt{0.05 	imes 0.95 / 200}}$$ and use NormalCDF to find the $$p-$$value.

Question 14

Which of the following is NOT a condition that must be checked before using the Central Limit Theorem for proportions?

Accepted Answer

The population must be normally distributed.

Question 15

If $$P(A) = 0.3$$, $$P(B) = 0.4$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A \cup B)$$?

Accepted Answer

$$0.3 + 0.4$$

Question 16

A 95% confidence interval for a mean is calculated as $$[45, 55]. $$Which of the following is a correct interpretation?

Accepted Answer

We are 95% confident that the true population mean is between 45 and 55.

Question 17

A researcher uses a 1-PropZTest on a TI-84 calculator. Which of the following inputs is required?

Accepted Answer

Sample size, number of successes, and hypothesized proportion.

Question 18

Which of the following best describes a Type I error?

Accepted Answer

Rejecting a true null hypothesis.

Question 19

A sample of 25 patients has a mean cholesterol level of 200 mg/dL and a sample standard deviation of 20 mg/dL. What is the standard error of the mean?

Accepted Answer

$$SE = \frac{20}{\sqrt{25}}$$

Question 20

A researcher wants to find the $$z-$$score that corresponds to the 90th percentile of a standard normal distribution. Which function and value should be used?

Accepted Answer

Use InvNorm with area = 0.90; $$z = 1.28$$

Probability, Normal Distribution, and Inference: Exam 2 Statistics Study Guide

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