Q: A researcher wants to estimate the mean IQ score of statistics students with 95% confidence and a margin of error of 3 IQ points. If $$\sigma = 15$$ and $$z_{\alpha/2} = 1.96$$, what is the minimum sample size required?

$$n = \left[ \dfrac{1.96 \times 15}{3} \right]^2 = 97$$

Q: What is the formula for the point estimate of the population mean $$\mu$$ from a confidence interval with upper and lower limits?

$$\mu = \dfrac{\text{upper confidence limit} + \text{lower confidence limit}}{2}$$

Question 1

Which of the following is the best point estimate for the population mean $$\mu$$ when using sample data?

Accepted Answer

The sample mean $$\bar{x}$$

Question 2

A researcher wants to construct a 95% confidence interval for the mean weight of apples in an orchard. The population standard deviation $$\sigma is $$known. Which formula should be used for the margin of error $$E$$?

Accepted Answer

$$E = z_{\alpha/2} \dfrac{\sigma}{\sqrt{n}}$$

Question 3

When is it appropriate to use the Student's t-distribution instead of the standard normal (z) distribution for constructing a confidence interval for a population mean?

Accepted Answer

When the population standard deviation $$\sigma is $$unknown and either the population is normal or $$n \u003e 30$$

Question 4

A sample of 6 randomly selected peanut butter cups has weights (in grams): 8.639, 8.689, 8.548, 8.980, 8.936, 9.042. What is the point estimate for the mean weight?

Accepted Answer

8.8057 g

Question 5

Given a sample size $$n = 6$$ and a 95% confidence level, what is the number of degrees of freedom used to find the critical value for the t-distribution?

Accepted Answer

5

Question 6

Which of the following statements about the Student's t-distribution is TRUE?

Accepted Answer

It has more variability (wider and flatter) than the standard normal distribution, especially for small samples

Question 7

A 95% confidence interval for the mean weight of peanut butter cups is calculated as $$8.5901 \u003c \mu \u003c 9.0213 g. $$What is the margin of error $$E$$?

Accepted Answer

$$E = \dfrac{9.0213 - 8.5901}{2} = 0.2156$$

Question 8

If the population standard deviation $$\sigma is $$unknown and the sample size $$n \u003c 30$$, which formula should be used for the margin of error $$E$$?

Accepted Answer

$$E = t_{\alpha/2} \dfrac{s}{\sqrt{n}}$$

Question 9

A researcher wants to estimate the mean IQ score of statistics students with 95% confidence and a margin of error of 3 IQ points. If $$\sigma = 15$$ and $$z_{\alpha/2} = 1.96$$, what is the minimum sample size required?

Accepted Answer

$$n = \left[ \dfrac{1.96 	imes 15}{3} ight]^2 = 97$$

Question 10

Which of the following is NOT a requirement for constructing a confidence interval for a population mean when $$\sigma is $$unknown?

Accepted Answer

The sample mean must equal the population mean

Question 11

What is the formula for the point estimate of the population mean $$\mu$$ from a confidence interval with upper and lower limits?

Accepted Answer

$$\mu = \dfrac{	ext{upper confidence limit} + 	ext{lower confidence limit}}{2}$$

Question 12

Why does the Student's t-distribution approach the standard normal distribution as sample size increases?

Accepted Answer

Because the standard deviation of the t-distribution decreases and approaches 1

Question 13

A confidence interval for a population mean is given by $$\bar{x} - E \u003c \mu \u003c \bar{x} + E. $$What does the value $$E$$ represent?

Accepted Answer

The margin of error

Question 14

If you calculate a required sample size $$n = 96.3 to $$estimate a population mean, what should you do according to the round-off rule?

Accepted Answer

Round up to $$n = 97$$

Question 15

Which of the following best describes the shape of the Student's t-distribution compared to the standard normal distribution for small sample sizes?

Accepted Answer

Wider and flatter

Question 16

A sample of 38 peanut butter cups has a total weight of 340.2 g. What is the expected mean weight per cup according to the package label?

Accepted Answer

$$8.953 g$$

Question 17

Which of the following is TRUE about degrees of freedom when constructing a confidence interval for a population mean using the t-distribution?

Accepted Answer

Degrees of freedom is equal to $$n - 1$$

Question 18

If the table for critical values does not include the exact number of degrees of freedom you need, what is a recommended approach?

Accepted Answer

Use the next lower number of degrees of freedom or interpolate

Question 19

A hospital administrator wants to estimate the average length of stay for patients. The population standard deviation is unknown, and the sample size is 25. What distribution should be used to construct the confidence interval?

Accepted Answer

Student's t-distribution

7.2

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