Q: For the president approval survey, is the sampling distribution of $$\hat{p}$$ approximately normal? Use $$n\hat{p}(1-\hat{p}) \geq 10.$$

Yes, because $$n\hat{p}(1-\hat{p}) = 1,200 \times 0.43 \times 0.57 = 294.12 \geq 10$$

Q: A sample of 64 candy bags has a mean weight of 4 ounces and a standard deviation of 0.2 ounces. What is the 92% confidence interval for the population mean?

$$4 \pm z_{0.04} \times \frac{0.2}{\sqrt{64}}$$

Q: A sample of 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?

$$\bar{x} \pm t_{0.025,24} \times \frac{s}{\sqrt{n}}$$

Q: In a survey of 1,000 adults, 865 own smartphones. What is the point estimate for the proportion and the standard error for a 94% confidence interval?

$$\hat{p} = 0.865$$, $$SE = \sqrt{\frac{0.865 \times 0.135}{1000}}$$

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?

$$4.98 \pm t_{0.02,11} \times 0.125$$

Question 1

Given a population with mean $$\mu = 3.19$$ and standard deviation $$\sigma = 0.15$$, what is the standard deviation of the sample mean for a sample of $$n = 25$$ gas stations?

Accepted Answer

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

Question 2

A survey of 1,200 people found that 43% approve of the president. What is the point estimate $$\hat{p}$$ and $$1 - \hat{p}$$?

Accepted Answer

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

Question 3

For the president approval survey, is the sampling distribution of $$\hat{p}$$ approximately normal? Use $$n\hat{p}(1-\hat{p}) \geq 10.$$

Accepted Answer

Yes, because $$n\hat{p}(1-\hat{p}) = 1,200 	imes 0.43 	imes 0.57 = 294.12 \geq 10$$

Question 4

A sample of 64 candy bags has a mean weight of 4 ounces and a standard deviation of 0.2 ounces. What is the 92% confidence interval for the population mean?

Accepted Answer

$$4 \pm z_{0.04} 	imes \frac{0.2}{\sqrt{64}}$$

Question 5

A sample of 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

$$\bar{x} \pm t_{0.025,24} 	imes \frac{s}{\sqrt{n}}$$

Question 6

In a survey of 1,000 adults, 865 own smartphones. What is the point estimate for the proportion and the standard error for a 94% confidence interval?

Accepted Answer

$$\hat{p} = 0.865$$, $$SE = \sqrt{\frac{0.865 	imes 0.135}{1000}}$$

Question 7

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?

Accepted Answer

$$4.98 \pm t_{0.02,11} 	imes 0.125$$

Question 8

A survey of 1,018 people found a sample mean of 14.5 books read and a standard error of 0.52. What is the standard deviation of the sample mean?

Accepted Answer

$$\sigma_{\bar{x}} = 0.52$$

Question 9

Interpret the result of a 95% confidence interval for the mean number of books read as (13.48, 15.53).

Accepted Answer

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

Question 10

Which formula is used to calculate the standard error of the sample mean for a sample size $$n$$ and population standard deviation $$\sigma$$?

Accepted Answer

$$SE = \frac{\sigma}{\sqrt{n}}$$

Question 11

If the confidence interval for the mean weight of candy bags is (3.95, 4.05) ounces at 92% confidence, what does this mean?

Accepted Answer

We are 92% confident that the true population mean weight is between 3.95 and 4.05 ounces.

Question 12

Which of the following is the correct formula for the margin of error in a confidence interval for a population proportion?

Accepted Answer

$$z_{\alpha/2} 	imes \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 13

A market research firm wants to estimate the proportion of adults with smartphones. Which condition must be met for the sampling distribution of $$\hat{p} to be $$approximately normal?

Accepted Answer

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

Question 14

If a sample size is less than or equal to 40 and the population standard deviation is unknown, which distribution should be used for confidence intervals?

Accepted Answer

The t-distribution

Question 15

A sample of 25 children has a mean of 6 months and a standard deviation of 3 months for using training wheels. What is the standard error of the mean?

Accepted Answer

$$SE = \frac{3}{\sqrt{25}} = 0.6$$

Question 16

Which of the following best describes the Central Limit Theorem?

Accepted Answer

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

Question 17

If the 90th percentile for the average gas price for 25 gas stations is $3.24, what does this mean?

Accepted Answer

90% of sample means are less than $3.24.

Question 18

Which formula is used to calculate the confidence interval for a population mean when the population standard deviation is unknown and $$n \leq 40$$?

Accepted Answer

$$\bar{x} \pm t_{\alpha/2, n-1} 	imes \frac{s}{\sqrt{n}}$$

MAT 118 Unit 4 Exam Review – Statistics Study Guidance

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