Q: A hospital administrator wants to estimate the proportion of patients who recover within 3 days. If $$p = 0.7$$ and $$n = 80$$, what is the standard error of the sample proportion?

$$\sigma_{\hat{p}} = \sqrt{\frac{0.7 \times 0.3}{80}} = 0.0576$$

Question 1

Which of the following best describes the main purpose of statistics in the context of sampling distributions?

Accepted Answer

To use a sample to make predictions or inferences about the entire population.

Question 2

Suppose the population mean ($$\mu) is 50 $$and the population standard deviation ($$\sigma) is 10. $$What is the standard error of the sample mean for a sample size of $$n = 25$$?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{10}{\sqrt{25}} = 2$$

Question 3

Which condition must be met for the sampling distribution of sample proportions to be approximately normal?

Accepted Answer

$$np \geq 15$$ and $$n(1-p) \geq 15$$

Question 4

If the population proportion of students who own a laptop is $$p = 0.8$$ and you take a sample of $$n = 50$$, what is the standard error of the sample proportion?

Accepted Answer

$$\sigma_{\hat{p}} = \sqrt{\frac{0.8(1-0.8)}{50}} = 0.0566$$

Question 5

A researcher collects a sample of $$n = 100$$ from a population with mean $$\mu = 60$$ and standard deviation $$\sigma = 20. $$According to the Central Limit Theorem, what is the shape of the sampling distribution of the sample mean?

Accepted Answer

Approximately normal, regardless of the population's shape.

Question 6

If you increase the sample size $$n in a $$study, what happens to the standard error of the sample mean?

Accepted Answer

It decreases.

Question 7

Suppose the population proportion of people who prefer tea over coffee is $$p = 0.4. $$You take a sample of $$n = 20. Is $$the sampling distribution of the sample proportion approximately normal?

Accepted Answer

No, because $$n(1-p) = 12 is $$less than 15.

Question 8

Which formula correctly represents the standard error of the sample mean?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$

Question 9

A population has $$\mu = 100$$ and $$\sigma = 15. $$What is the probability that a sample mean from $$n = 36 is $$less than 95? (Assume normality.)

Accepted Answer

First, calculate $$z = \frac{95 - 100}{15/\sqrt{36}} = \frac{-5}{2.5} = -2. $$The probability is approximately 0.0228.

Question 10

Which statement is TRUE about the mean of the sampling distribution of the sample mean?

Accepted Answer

It is equal to the population mean $$\mu.$$

Question 11

A hospital administrator wants to estimate the proportion of patients who recover within 3 days. If $$p = 0.7$$ and $$n = 80$$, what is the standard error of the sample proportion?

Accepted Answer

$$\sigma_{\hat{p}} = \sqrt{\frac{0.7 	imes 0.3}{80}} = 0.0576$$

Question 12

If a sample mean is 2 standard errors away from the population mean, how would you describe this sample mean?

Accepted Answer

It is somewhat unusual but not extremely rare.

Question 13

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

Accepted Answer

The population standard deviation must be known.

Question 14

A company finds that 36% of its customers use a mobile app. If a random sample of 100 customers is taken, what is the probability that the sample proportion is between 0.30 and 0.50? (Assume normality.)

Accepted Answer

Calculate $$z$$ for 0.30: $$z = \frac{0.30 - 0.36}{\sqrt{0.36 	imes 0.64 / 100}} = -0.75$$; for 0.50: $$z = \frac{0.50 - 0.36}{\sqrt{0.36 	imes 0.64 / 100}} = 1.75. $$Probability is $$P(-0.75 \u003c z \u003c 1.75) = 0.9599 - 0.2266 = 0.7333.$$

Question 15

Which of the following is TRUE about the standard error of the sample proportion as sample size increases?

Accepted Answer

It decreases.

Question 16

If the population mean commute time is 45 minutes with a standard deviation of 12 minutes, and a sample of $$n = 36 is $$taken, what value would 25% of sample means fall below? (Assume normality.)

Accepted Answer

Find $$z$$ for 0.25: $$z = -0.67$$. $$x = 45 + (-0.67) 	imes \frac{12}{6} = 45 - 1.34 = 43.66$$ minutes.

Question 17

Which of the following best describes a sampling distribution?

Accepted Answer

It is the distribution of a statistic (like the mean or proportion) from all possible samples of a given size from the population.

Question 18

A sample mean is found to be 3.5 when the population mean is 3.5. What does this indicate about the sample?

Accepted Answer

The sample mean is an unbiased estimator of the population mean.

Chapter 7 – Sampling Distributions (STAT 201: Elementary Statistics)

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