Question 1

A researcher wants to estimate the average weight of apples in an orchard. She collects a simple random sample of 40 apples and finds a sample mean of 150g and a sample standard deviation of 20g. Construct a 95% confidence interval for the population mean weight of apples. Which formula should she use for the margin of error?

Accepted Answer

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

Question 2

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

Accepted Answer

The population standard deviation $$\sigma$$ must be known.

Question 3

A 95% confidence interval for the mean height of students in a class is (165.6 cm, 174.6 cm). What does this interval mean?

Accepted Answer

We are 95% confident that the true mean height of the population falls between 165.6 cm and 174.6 cm.

Question 4

When constructing a confidence interval for a population mean with $$\sigma$$ unknown and $$n = 25$$, which distribution should be used to find the critical value?

Accepted Answer

Student t distribution with $$df = 24$$

Question 5

A sample of 6 Reese's Peanut Butter Cups Miniatures has a mean weight of 8.8057 g and a sample standard deviation of 0.2054 g. Using a 95% confidence level, the critical value from the t-distribution is 2.571. What is the margin of error?

Accepted Answer

$$E = 2.571 \cdot \dfrac{0.2054}{\sqrt{6}} = 0.2156$$

Question 6

Suppose you want to estimate the mean IQ score of statistics students with 95% confidence and a margin of error of 3 points. If the population standard deviation is estimated to be 15, what is the minimum sample size required?

Accepted Answer

$$n = \left( \dfrac{1.96 \cdot 15}{3} \$$right)^2$$ = 97$$

Question 7

If the population standard deviation $$\sigma is $$known, which formula should be used for the margin of error when constructing a confidence interval for the mean?

Accepted Answer

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

Question 8

A sample of 15 birth weights of girls has a sample mean of 30.9 hg and a known population standard deviation of 2.9 hg. What is the 95% confidence interval for the population mean?

Accepted Answer

$$30.9 \pm 1.96 \cdot \dfrac{2.9}{\sqrt{15}}$$

Question 9

Which of the following statements about the Student t distribution is TRUE?

Accepted Answer

As the sample size increases, the Student t distribution approaches the standard normal distribution.

Question 10

You are given a confidence interval for a population mean as (8.5901, 9.0213). What is the point estimate and margin of error?

Accepted Answer

Point estimate: $$\dfrac{8.5901 + 9.0213}{2} = 8.8057$$; Margin of error: $$\dfrac{9.0213 - 8.5901}{2} = 0.2156$$

Question 11

A researcher wants to determine the sample size needed to estimate a population mean with a margin of error $$E$$ and a known population standard deviation $$\sigma. $$Which formula should she use?

Accepted Answer

$$n = \left( \dfrac{z_{\alpha/2} \cdot \sigma}{E} \$$right)^2$$

Question 12

If you do not know the population standard deviation $$\sigma$$ and have a small sample size ($$n \u003c 30$$), but the population is not normally distributed, what is the recommended method for constructing a confidence interval for the mean?

Accepted Answer

Use the bootstrapping method or a nonparametric method.

Question 13

Which of the following best describes 'degrees of freedom' when constructing a confidence interval for a mean with unknown $$\sigma$$?

Accepted Answer

The number of sample values that can vary after certain restrictions have been imposed, calculated as $$n - 1.$$

Question 14

A company wants to estimate the average time (in minutes) it takes to assemble a product. They want to be 99% confident that their estimate is within 2 minutes of the true mean. If the standard deviation is estimated to be 6 minutes, what is the minimum sample size required?

Accepted Answer

$$n = \left( \dfrac{2.576 \cdot 6}{2} \$$right)^2$$ = 59.7 \approx 60$$

Question 15

When rounding the limits of a confidence interval constructed from summary statistics (mean, standard deviation, sample size), what is the correct rounding rule?

Accepted Answer

Round the confidence interval limits to the same number of decimal places as the sample mean.

Question 16

A sample of 50 patients has a mean cholesterol level of 200 mg/dL and a sample standard deviation of 25 mg/dL. The population is not known to be normally distributed. Which method is most appropriate for constructing a 95% confidence interval for the mean cholesterol level?

Accepted Answer

Use the Student t distribution with $$df = 49$$ because $$n \u003e 30.$$

Question 17

If you are using the range rule of thumb to estimate the standard deviation for sample size calculations, which formula should you use?

Accepted Answer

$$\sigma \approx \dfrac{	ext{range}}{4}$$

Estimating Parameters and Determining Sample Sizes: Estimating a Population Mean

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