Q: Suppose the true proportion of patients experiencing post-surgical complications after eye laser surgery is $$p = 0.13. If a $$random sample of $$n = 1000$$ patients is taken, what is the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

$$\sqrt{\dfrac{0.13 \times 0.87}{1000}}$$

Question 1

Which of the following best describes the difference between a parameter and a statistic?

Accepted Answer

A parameter is a fixed but unknown value describing a population, while a statistic is a random variable calculated from a sample.

Question 2

Suppose the true proportion of patients experiencing post-surgical complications after eye laser surgery is $$p = 0.13. If a $$random sample of $$n = 1000$$ patients is taken, what is the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\sqrt{\dfrac{0.13 	imes 0.87}{1000}}$$

Question 3

Which condition must be met for the normal approximation to the sampling distribution of $$\hat{p} to be $$valid?

Accepted Answer

Both $$np \geq 10$$ and $$n(1-p) \geq 10$$ must be satisfied.

Question 4

A bottling company fills bottles with cola. The contents are normally distributed with mean $$298 mL $$and standard deviation $$3 mL. $$What is the probability that a randomly selected bottle contains less than $$295 mL$$?

Accepted Answer

Find $$P(X \u003c 295)$$ where $$X \sim N(298, 3)$$; $$z = \dfrac{295 - 298}{3} = -1$$; so $$P(X \u003c 295) \approx 0.1587$$

Question 5

A manufacturer claims the mean battery life is $$54$$ months with a standard deviation of $$6$$ months. If a consumer group tests a random sample of $$n = 50$$ batteries, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\dfrac{6}{\sqrt{50}}$$

Question 6

Which statement about the Central Limit Theorem (CLT) is correct?

Accepted Answer

The CLT states that for sufficiently large samples, the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution.

Question 7

If the population distribution is normal, what is true about the sampling distribution of the sample mean for any sample size $$n$$?

Accepted Answer

It is exactly normal with mean $$\mu$$ and standard deviation $$\dfrac{\sigma}{\sqrt{n}}$$ for any $$n.$$

Question 8

A sample of $$n = 6$$ bottles is taken from a bottling process where the contents are normally distributed with mean $$298 mL $$and standard deviation $$3 mL. $$What is the probability that the mean content of the six bottles is less than $$295 mL$$?

Accepted Answer

Calculate $$z = \dfrac{295 - 298}{3/\sqrt{6}} = \dfrac{-3}{1.225} \approx -2.45$$; $$P(\bar{x} \u003c 295) \approx 0.0071$$

Question 9

In the context of sampling distributions, what does it mean for a statistic to be a random variable?

Accepted Answer

Its value can vary depending on which individuals are included in the sample.

Question 10

A researcher draws a random sample of $$n = 1000$$ patients and finds that $$25$$ experienced post-surgical complications. The known population proportion is $$p = 0.013. Is $$this sample result compatible with the population proportion?

Accepted Answer

No, because $$25/1000 = 0.025 is $$much greater than $$0.013$$ and is highly unlikely under the sampling distribution.

Question 11

Which of the following is NOT an assumption required for the validity of the normal approximation to the sampling distribution of the sample mean?

Accepted Answer

The population mean must be known.

Question 12

If the sample size $$n$$ increases, what happens to the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

It decreases.

Question 13

A population has mean $$\mu = 75.8$$ and standard deviation $$\sigma = 8.84. $$What is the standard deviation of the sampling distribution of the mean for samples of size $$n = 25$$?

Accepted Answer

$$\dfrac{8.84}{\sqrt{25}} = 1.768$$

Question 14

Which of the following best describes the center of the sampling distribution of the sample mean?

Accepted Answer

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

Question 15

A population proportion $$p is $$estimated using a sample of size $$n. $$Which formula gives the standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$\sqrt{\dfrac{p(1-p)}{n}}$$

Question 16

Why is it important that the sample size is no greater than 10% of the population size when sampling without replacement?

Accepted Answer

To maintain the independence of sampled individuals.

Question 17

A sample of $$n = 50$$ batteries is tested, and the mean lifetime is found to be $$52$$ months. The population mean is $$54$$ months and standard deviation is $$6$$ months. What is the $$z-$$score for this sample mean?

Accepted Answer

$$z = \dfrac{52 - 54}{6/\sqrt{50}} = -2.36$$

Question 18

Which of the following is a real-world example of a statistic?

Accepted Answer

The average score of a random sample of 30 students on a final exam.

Chapter 14: Sampling Distribution Models – Study Notes

Study Guide - Practice Questions

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