Q: A sample of 12 rain dates has a mean pH of 4.98 and a standard error of 0.125. What is the 96% confidence interval for the population mean pH?

$$ 4.98 \pm t^* \times 0.125 $$

Question 1

A random sample of 25 gas stations in Syracuse is taken, where the population mean price of unleaded gasoline is $$3.19 and the population standard deviation is 0.15. $$What is the standard deviation of the sample mean?

Accepted Answer

$$ \sigma_{\bar{x}} = \frac{0.15}{\sqrt{25}} = 0.03 $$

Question 2

Given a sample of 1,200 people, 43% feel the president is doing an acceptable job. What is the point estimate $$ \hat{p} $$ and $$ 1 - \hat{p} $$?

Accepted Answer

$$ \hat{p} = 0.43, 1 - \hat{p} = 0.57 $$

Question 3

A sample of 64 bags of candy has a mean weight of 4 ounces and a standard deviation of 0.2 ounces. Construct a 92% confidence interval for the population mean. Which formula should be used?

Accepted Answer

$$ \bar{x} \pm z^* \frac{\sigma}{\sqrt{n}} $$

Question 4

Suppose 25 children used training wheels for an average of 6 months with a sample standard deviation of 3 months. Which formula is appropriate for constructing a 95% confidence interval for the population mean?

Accepted Answer

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

Question 5

A market research firm surveys 1,000 adults and finds 865 own smartphones. What is the point estimate for the proportion of adults with smartphones?

Accepted Answer

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

Question 6

For the smartphone survey, which formula is used to calculate the standard error for the proportion?

Accepted Answer

$$ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$

Question 7

A sample of 12 rain dates has a mean pH of 4.98 and a standard error of 0.125. What is the 96% confidence interval for the population mean pH?

Accepted Answer

$$ 4.98 \pm t^* 	imes 0.125 $$

Question 8

A survey of 1,018 people finds a sample mean of 14.5 books read and a standard deviation of 16.6 books. What is the standard error of the mean?

Accepted Answer

$$ SE = \frac{16.6}{\sqrt{1018}} \approx 0.52 $$

Question 9

Which of the following best describes the Central Limit Theorem?

Accepted Answer

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

Question 10

If a confidence interval for the mean weight of candy bags is (3.96, 4.04) ounces at 92% confidence, what does this mean?

Accepted Answer

We are 92% confident that the true population mean weight is between 3.96 and 4.04 ounces.

Question 11

A 95% confidence interval for the mean number of books read is (13.48, 15.53). What is the correct interpretation?

Accepted Answer

We are 95% confident that the true population mean number of books read is between 13.48 and 15.53.

Question 12

Which formula is used to check if the sampling distribution of the sample proportion is approximately normal?

Accepted Answer

$$ n\hat{p}(1-\hat{p}) \geq 10 $$

Question 13

If the sample size is small ($$ n \leq 40 ) $$and the population standard deviation is unknown, which distribution should be used for confidence intervals?

Accepted Answer

The t-distribution

Question 14

A survey finds a sample proportion $$ \hat{p} = 0.43 $$ with $$ n = 1200 . $$What is the standard deviation of the sample proportion?

Accepted Answer

$$ \sqrt{\frac{0.43 	imes 0.57}{1200}} \approx 0.0142 $$

Question 15

Which scenario requires the use of the t-distribution instead of the z-distribution?

Accepted Answer

Sample size is small and population standard deviation is unknown.

Question 16

If the confidence level increases from 92% to 99%, what happens to the width of the confidence interval?

Accepted Answer

The confidence interval becomes wider.

Question 17

Which of the following is NOT a requirement for constructing a confidence interval for a population mean?

Accepted Answer

The sample mean must equal the population mean.

Question 18

A sample of 25 gas stations has a mean price of $$3.19 and a standard deviation of 0.15. $$What is the probability that the sample mean is less than $3.16?

Accepted Answer

Calculate $$ z = \frac{3.16 - 3.19}{0.03} = -1 $$; probability is approximately 0.1587.

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

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