Q: For the survey above, what is the standard deviation of the sampling distribution of $$\hat{p}$$? Use $$\hat{p} = 0.43$$ and $$n = 1200.$$

$$\sqrt{\frac{0.43 \times 0.57}{1200}} \approx 0.0143$$

Q: A sample of 64 small bags of candies has a mean weight of 4 ounces and a standard deviation of 0.2 ounces. Construct a 92% confidence interval for the population mean. What is the margin of error? (Use $$z^* = 1.75$$ for 92% confidence.)

$$1.75 \times \frac{0.2}{\sqrt{64}} = 0.04375$$

Q: For the smartphone survey above, what is the standard error for the proportion estimate?

$$\sqrt{\frac{0.865 \times 0.135}{1000}} \approx 0.0108$$

Q: A random sample of 12 rain dates has a sample mean pH of 4.98 and a standard error of 0.125. What is the 96% confidence interval for the population mean pH? (Use $$t^* = 2.201$$ for 11 degrees of freedom.)

$$4.98 \pm 2.201 \times 0.125$$

Question 1

Suppose the cost of unleaded gasoline in Syracuse follows an unknown distribution with a mean of $$3.19 and a standard deviation of 0.15. If 25 $$gas stations are randomly chosen, what is the standard deviation of the sample mean?

Accepted Answer

$$\frac{0.15}{\sqrt{25}} = 0.03$$

Question 2

A survey of 1,200 people found that 43% feel the president is doing an acceptable job. What is the point estimate $$\hat{p}$$ for the population proportion?

Accepted Answer

$$\hat{p} = 0.43$$

Question 3

For the survey above, what is the standard deviation of the sampling distribution of $$\hat{p}$$? Use $$\hat{p} = 0.43$$ and $$n = 1200.$$

Accepted Answer

$$\sqrt{\frac{0.43 	imes 0.57}{1200}} \approx 0.0143$$

Question 4

A sample of 64 small bags of candies has a mean weight of 4 ounces and a standard deviation of 0.2 ounces. Construct a 92% confidence interval for the population mean. What is the margin of error? (Use $$z^* = 1.75$$ for 92% confidence.)

Accepted Answer

$$1.75 	imes \frac{0.2}{\sqrt{64}} = 0.04375$$

Question 5

Suppose 25 children used training wheels for an average of 6 months with a sample standard deviation of 3 months. What formula should be used to construct a 95% confidence interval for the population mean?

Accepted Answer

$$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$$

Question 6

A market research firm surveyed 1,000 adults and found that 865 own smartphones. What is the point estimate for the proportion of adults in the city who own smartphones?

Accepted Answer

$$\hat{p} = \frac{865}{1000} = 0.865$$

Question 7

For the smartphone survey above, what is the standard error for the proportion estimate?

Accepted Answer

$$\sqrt{\frac{0.865 	imes 0.135}{1000}} \approx 0.0108$$

Question 8

A random sample of 12 rain dates has a sample mean pH of 4.98 and a standard error of 0.125. What is the 96% confidence interval for the population mean pH? (Use $$t^* = 2.201$$ for 11 degrees of freedom.)

Accepted Answer

$$4.98 \pm 2.201 	imes 0.125$$

Question 9

A survey of 1,018 people found a sample mean of 14.5 books read in the past year with a standard deviation of 16.6 books. What is the standard error of the mean?

Accepted Answer

$$\frac{16.6}{\sqrt{1018}} \approx 0.520$$

Question 10

Which of the following best describes the interpretation of a 95% confidence interval for the mean number of books read?

Accepted Answer

We are 95% confident that the true population mean lies within the interval calculated from the sample.

Question 11

If the sample size increases, what happens to the standard error of the mean?

Accepted Answer

It decreases, because $$\frac{\sigma}{\sqrt{n}}$$ gets smaller as $$n$$ increases.

Question 12

Which formula is used to calculate the probability that the sample mean falls within a certain range for a normal distribution?

Accepted Answer

$$z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$$

Question 13

A confidence interval for a population proportion is calculated as $$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. $$What does $$z^*$$ represent?

Accepted Answer

The critical value from the standard normal distribution for the desired confidence level.

Question 14

Which of the following is a requirement for using the normal approximation for the sampling distribution of a proportion?

Accepted Answer

$$np(1-p) \geq 10$$

Question 15

If a sample mean is much higher than the population mean, which of the following could explain this result?

Accepted Answer

Sampling variability or a non-representative sample.

Question 16

Which formula is used to calculate the margin of error for a confidence interval for the mean when the population standard deviation is unknown?

Accepted Answer

$$t^* \frac{s}{\sqrt{n}}$$

Question 17

A 90th percentile for the average gas price for 25 gas stations is calculated. What does this value represent?

Accepted Answer

The value below which 90% of sample means would fall.

Question 18

Which of the following best describes the shape of the sampling distribution of the sample mean for a large sample size, regardless of the population distribution?

Accepted Answer

Approximately normal, due to the Central Limit Theorem.

Unit 4 Exam Review – Sampling Distributions, Confidence Intervals, and Probability

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