Q: A pharmaceutical company wants to estimate the average time it takes for a new drug to take effect. If the population standard deviation is known to be 8 minutes, and they want a 95% confidence interval with a margin of error no greater than 2 minutes, what is the minimum sample size required? Use $$z_{\alpha/2} = 1.96$$ for 95% confidence.

$$n = \left( \frac{1.96 \times 8}{2} \$$right)^2$$ = 61.47 \Rightarrow 62$$

Q: A researcher collects a random sample of 50 patients and finds a sample mean blood pressure of 120 mmHg with a population standard deviation of 15 mmHg. What is the 90% confidence interval for the population mean? Use $$z_{\alpha/2} = 1.645.$$

$$120 \pm 1.645 \times \frac{15}{\sqrt{50}} \Rightarrow (116.51, 123.49)$$

Q: A hospital wants to estimate the average recovery time for a surgery. They collect a sample of 25 patients, with a mean recovery time of 10 days and a sample standard deviation of 4 days. What is the 95% confidence interval for the population mean recovery time? Use $$t_{\alpha/2,24} = 2.064.$$

$$10 \pm 2.064 \times \frac{4}{\sqrt{25}} \Rightarrow (9.35, 10.65)$$

Q: Which of the following is a correct formula for the margin of error in a confidence interval for a population mean when the population standard deviation is known?

$$E = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$$

Q: A nutritionist wants to estimate the mean calorie intake of adults in a city. She wants to be 99% confident that her estimate is within 50 calories of the true mean. If the population standard deviation is 200 calories, what sample size should she use? Use $$z_{\alpha/2} = 2.576.$$

$$n = \left( \frac{2.576 \times 200}{50} \$$right)^2$$ = 106.02 \Rightarrow 107$$

Question 1

Which of the following statements best describes the Central Limit Theorem?

Accepted Answer

It states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Question 2

A pharmaceutical company wants to estimate the average time it takes for a new drug to take effect. If the population standard deviation is known to be 8 minutes, and they want a 95% confidence interval with a margin of error no greater than 2 minutes, what is the minimum sample size required? Use $$z_{\alpha/2} = 1.96$$ for 95% confidence.

Accepted Answer

$$n = \left( \frac{1.96 	imes 8}{2} \$$right)^2$$ = 61.47 \Rightarrow 62$$

Question 3

When is it appropriate to use the normal distribution to approximate the distribution of sample means?

Accepted Answer

When the sample size is large (typically $$n \geq 30$$), regardless of the population's distribution.

Question 4

A researcher collects a random sample of 50 patients and finds a sample mean blood pressure of 120 mmHg with a population standard deviation of 15 mmHg. What is the 90% confidence interval for the population mean? Use $$z_{\alpha/2} = 1.645.$$

Accepted Answer

$$120 \pm 1.645 	imes \frac{15}{\sqrt{50}} \Rightarrow (116.51, 123.49)$$

Question 5

Which property is NOT true about confidence intervals for population means?

Accepted Answer

The confidence interval always contains the population mean.

Question 6

A random sample of 36 students has a mean test score of 78 with a population standard deviation of 12. What is the probability that the sample mean is greater than 80? Assume the population mean is 75. Use the Central Limit Theorem.

Accepted Answer

First, $$\mu = 75$$, $$\sigma = 12$$, $$n = 36$$. $$\sigma_{\bar{x}} = \frac{12}{\sqrt{36}} = 2$$. $$z = \frac{80 - 75}{2} = 2.5$$. $$P(\bar{x} \u003e 80) = P(z \u003e 2.5) = 0.0062$$

Question 7

Which of the following best explains why the Central Limit Theorem is important in statistics?

Accepted Answer

It allows us to use the normal distribution to make inferences about sample means, even when the population distribution is not normal.

Question 8

A hospital wants to estimate the average recovery time for a surgery. They collect a sample of 25 patients, with a mean recovery time of 10 days and a sample standard deviation of 4 days. What is the 95% confidence interval for the population mean recovery time? Use $$t_{\alpha/2,24} = 2.064.$$

Accepted Answer

$$10 \pm 2.064 	imes \frac{4}{\sqrt{25}} \Rightarrow (9.35, 10.65)$$

Question 9

Which of the following is a correct formula for the margin of error in a confidence interval for a population mean when the population standard deviation is known?

Accepted Answer

$$E = z_{\alpha/2} 	imes \frac{\sigma}{\sqrt{n}}$$

Question 10

A nutritionist wants to estimate the mean calorie intake of adults in a city. She wants to be 99% confident that her estimate is within 50 calories of the true mean. If the population standard deviation is 200 calories, what sample size should she use? Use $$z_{\alpha/2} = 2.576.$$

Accepted Answer

$$n = \left( \frac{2.576 	imes 200}{50} \$$right)^2$$ = 106.02 \Rightarrow 107$$

Question 11

Which of the following is NOT a requirement for using the Central Limit Theorem to approximate probabilities for sample means?

Accepted Answer

The population standard deviation must be known.

Question 12

A random sample of 100 light bulbs has a mean lifetime of 1200 hours. The population standard deviation is 100 hours. What is the probability that the sample mean is between 1180 and 1220 hours if the population mean is 1200 hours?

Accepted Answer

First, $$\sigma_{\bar{x}} = \frac{100}{\sqrt{100}} = 10$$. $$z_1$$ = \frac{1180 - 1200}{10} = -2$$, z_2$ = \frac{1220 - 1200}{10} = 2$$. $$P(-2 \u003c z \u003c 2) = 0.9544$$

Question 13

Which of the following best describes a confidence interval?

Accepted Answer

A range of values, calculated from sample data, that is likely to contain the population mean with a specified probability.

Question 14

A random sample of 64 batteries has a mean lifetime of 90 hours. The population standard deviation is 16 hours. What is the 99% confidence interval for the population mean? Use $$z_{\alpha/2} = 2.576.$$

Accepted Answer

$$90 \pm 2.576 	imes \frac{16}{\sqrt{64}} \Rightarrow (84.85, 95.15)$$

Question 15

Which of the following is a correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

We are 95% confident that the interval contains the true population mean.

Question 16

A random sample of 40 patients has a mean cholesterol level of 200 mg/dL. The population standard deviation is 20 mg/dL. What is the probability that the sample mean is less than 195 mg/dL if the population mean is 205 mg/dL?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{20}{\sqrt{40}} = 3.16$$. $$z = \frac{195 - 205}{3.16} = -3.16$$. $$P(z \u003c -3.16) = 0.0008$$

Question 17

Which of the following is NOT a property of the sampling distribution of the sample mean?

Accepted Answer

The sampling distribution is always normal regardless of sample size.

Step-by-Step Guidance for Central Limit Theorem, Normal Distribution, and Confidence Intervals

Study Guide - Practice Questions

Study Guide - Flashcards