Q: A pharmaceutical company wants to estimate the proportion of patients who experience side effects from a new drug. They survey a random sample of 400 patients and find that 60 experienced side effects. Construct a 95% confidence interval for the true proportion of patients who experience side effects. Which of the following is the correct interval?

$$0.15 \pm 1.96 \times \sqrt{\frac{0.15 \times 0.85}{400}}$$

Q: A researcher wants to test if a new teaching method is more effective than the traditional method. Which of the following is the correct null hypothesis?

$$H_0$$: p_{\text{new}} = p_{\text{traditional}}$$

Q: A sample of 100 light bulbs has a mean lifetime of 1200 hours and a standard deviation of 100 hours. What is the 95% confidence interval for the mean lifetime?

$$1200 \pm 1.96 \times \frac{100}{\sqrt{100}}$$

Question 1

A pharmaceutical company wants to estimate the proportion of patients who experience side effects from a new drug. They survey a random sample of 400 patients and find that 60 experienced side effects. Construct a 95% confidence interval for the true proportion of patients who experience side effects. Which of the following is the correct interval?

Accepted Answer

$$0.15 \pm 1.96 	imes \sqrt{\frac{0.15 	imes 0.85}{400}}$$

Question 2

Which of the following statements best describes the Central Limit Theorem (CLT)?

Accepted Answer

As the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the population's distribution.

Question 3

A survey is conducted by calling landline phone numbers listed in a directory. Which type of bias is most likely to affect the results?

Accepted Answer

Sampling bias

Question 4

A fair coin is flipped 10 times. What is the probability of getting exactly 7 heads? (Use the binomial formula.)

Accepted Answer

$$\binom{10}{7} (0.5)^7$ (0.5)^3$

Question 5

Which of the following is a parameter, not a statistic?

Accepted Answer

The mean height of all adults in the United States

Question 6

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the probability that a randomly selected value is between 85 and 115?

Accepted Answer

Approximately 68%

Question 7

A researcher wants to test if a new teaching method is more effective than the traditional method. Which of the following is the correct null hypothesis?

Accepted Answer

$$H_0$$: p_{	ext{new}} = p_{	ext{traditional}}$$

Question 8

A sample of 50 students has a mean test score of 78 with a standard deviation of 10. What is the standard error of the mean?

Accepted Answer

$$\frac{10}{\sqrt{50}}$$

Question 9

Which of the following best describes a Bernoulli trial?

Accepted Answer

A random experiment with exactly two possible outcomes

Question 10

A polling organization reports a margin of error of 3% for a 95% confidence interval. What does this mean?

Accepted Answer

The true population proportion is likely within 3% of the sample proportion, 95% of the time.

Question 11

A hospital administrator wants to know if the proportion of patients who recover within a week is different between two treatments. Which statistical test should be used?

Accepted Answer

Two-sample proportion z-test

Question 12

If the p-value for a hypothesis test is 0.03 and the significance level is 0.05, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis

Question 13

Which of the following is NOT a requirement for using the normal approximation to the binomial distribution?

Accepted Answer

The population must be normally distributed

Question 14

A sample of 200 voters shows that 120 support a new policy. What is the sample proportion?

Accepted Answer

$$\frac{120}{200}$$

Question 15

Which of the following best describes measurement bias?

Accepted Answer

A systematic error in data collection that skews results in a particular direction

Question 16

A normal distribution has mean $$\mu = 50$$ and standard deviation $$\sigma = 5. $$What z-score corresponds to a value of 60?

Accepted Answer

$$z = \frac{60 - 50}{5}$$

Question 17

Which of the following is the correct formula for the standard deviation of a binomial distribution with parameters $$n$$ and $$p$$?

Accepted Answer

$$\sqrt{np(1-p)}$$

Question 18

A sample of 100 light bulbs has a mean lifetime of 1200 hours and a standard deviation of 100 hours. What is the 95% confidence interval for the mean lifetime?

Accepted Answer

$$1200 \pm 1.96 	imes \frac{100}{\sqrt{100}}$$

Question 19

In a hypothesis test, what does the p-value represent?

Accepted Answer

The probability of observing a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true.

Question 20

A researcher collects a sample of size $$n = 36$$ from a population with mean $$\mu = 80$$ and standard deviation $$\sigma = 12. $$According to the Central Limit Theorem, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\frac{12}{\sqrt{36}}$$

Midterm #2 Review Guide: Probability, Distributions, Sampling, and Inference

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