Question 1

Which of the following is a requirement for constructing a confidence interval to estimate a population mean when the population standard deviation $$\sigma is $$not known?

Accepted Answer

The sample must be a simple random sample.

Question 2

What is the formula for the margin of error $$E$$ when constructing a confidence interval for a population mean $$\mu$$ with unknown $$\sigma$$?

Accepted Answer

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

Question 3

A researcher collects a simple random sample of size $$n = 25$$ from a population with unknown $$\sigma. $$The sample mean is $$50$$ and the sample standard deviation is $$8. $$What is the degrees of freedom for the $$t$$ distribution used to construct a confidence interval?

Accepted Answer

$$24$$

Question 4

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

Accepted Answer

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

Question 5

A sample of $$n = 6$$ peanut butter cups has a sample mean weight of $$8.81 g $$and a sample standard deviation of $$0.20 g. $$Construct a 95% confidence interval for the population mean weight. The critical value is $$t_{0.025, 5} = 2.571. $$What is the margin of error?

Accepted Answer

$$E = 2.571 \cdot \dfrac{0.20}{\sqrt{6}} = 0.21$$

Question 6

Why did William Gosset use the pseudonym 'Student' when publishing his work on the $$t$$ distribution?

Accepted Answer

His employer, Guinness Brewery, did not want the research results published under his real name.

Question 7

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

Accepted Answer

$$n = \left( \dfrac{1.96 \cdot 15}{3} \$$right)^2$$ = 96.04$$, so $$n = 97$$

Question 8

If a 95% confidence interval for the mean weight of peanut butter cups is $$8.5901 \u003c \mu \u003c 9.0213$$, what is the point estimate for the mean?

Accepted Answer

$$\dfrac{8.5901 + 9.0213}{2} = 8.8057$$

Question 9

Which of the following is a conservative approach when estimating the required sample size for a confidence interval?

Accepted Answer

Use a value for $$\sigma$$ that is larger than the true value.

Question 10

A sample of $$n = 40 is $$drawn from a population with unknown $$\sigma. $$The sample mean is $$100$$ and the sample standard deviation is $$10. $$Which distribution should be used to construct a 95% confidence interval for the population mean?

Accepted Answer

Student $$t$$ distribution with $$39$$ degrees of freedom

Question 11

Which of the following best describes the effect of increasing the sample size on the width of a confidence interval for the mean?

Accepted Answer

The confidence interval becomes narrower.

Question 12

When constructing a confidence interval for a population mean with $$n \u003c 30$$, which of the following is NOT a requirement?

Accepted Answer

The sample must have more than one mode.

Question 13

If the sample size is increased but the confidence level remains the same, what happens to the margin of error?

Accepted Answer

The margin of error decreases.

Question 14

Which of the following is the correct formula for estimating the required sample size $$n to $$estimate a population mean with a specified margin of error $$E$$, confidence level $$z_{\alpha/2}$$, and estimated standard deviation $$\sigma$$?

Accepted Answer

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

Question 15

A sample of $$n = 12 is $$drawn from a population with unknown $$\sigma. $$Which of the following statements about the $$t$$ distribution is correct?

Accepted Answer

The $$t$$ distribution will be wider and have heavier tails than the standard normal distribution.

Question 16

If the sample standard deviation $$s is $$used as an estimate for $$\sigma in $$the sample size formula, what should be done as more data are collected?

Accepted Answer

Update the estimate of $$\sigma$$ and refine the sample size calculation.

7.2 Estimating Population Means and Determining Sample Sizes: Confidence Intervals and the Student t Distribution

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