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

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

Question 1

Which of the following best describes the main goal of constructing a confidence interval for a population mean $$\mu$$?

Accepted Answer

To use a sample mean $$\bar{x} to $$make an inference about the value of the corresponding population mean $$\mu.$$

Question 2

Which formula is used to calculate the test statistic $$t$$ when estimating a population mean with unknown $$\sigma$$?

Accepted Answer

$$t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$$

Question 3

A researcher wants to estimate the mean birth weight of newborn girls. She collects a simple random sample of $$n = 15$$ and finds $$\bar{x} = 30.9 hg $$and $$s = 2.9 hg. $$What is the number of degrees of freedom for constructing a confidence interval?

Accepted Answer

$$df = 14$$

Question 4

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

Accepted Answer

It has greater variability than the standard normal distribution, especially for small sample sizes.

Question 5

When constructing a confidence interval for a population mean with unknown $$\sigma$$, which condition(s) must be satisfied?

Accepted Answer

The sample is a simple random sample and the population is normally distributed or $$n \u003e 30.$$

Question 6

A 95% confidence interval for a population mean is calculated as $$29.4 \leq \mu \leq 32.4. $$What is the point estimate for $$\mu$$ and the margin of error $$E$$?

Accepted Answer

Point estimate: $$30.9$$, Margin of error: $$1.5$$

Question 7

Which Excel function would you use to find the margin of error for a confidence interval for the mean when $$\sigma is $$unknown?

Accepted Answer

=CONFIDENCE.T(alpha, standard_dev, size)

Question 8

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

Accepted Answer

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

Question 9

If the population standard deviation $$\sigma is $$known, which distribution should be used to construct a confidence interval for the mean?

Accepted Answer

Standard normal ($$z) $$distribution

Question 10

A sample of $$n = 15$$ birth weights has $$\bar{x} = 30.9 hg $$and $$\sigma = 2.9 hg. $$What is the margin of error $$E$$ for a 95% confidence interval?

Accepted Answer

$$E = 1.47$$

Question 11

Which of the following is the correct formula for the required sample size $$n to $$estimate a population mean with a specified margin of error $$E$$ and known $$\sigma$$?

Accepted Answer

$$n = \left[ \frac{z_{\alpha/2} \sigma}{E} \right]^2$$

Question 12

Why does the Student $$t$$ distribution have heavier tails than the standard normal distribution?

Accepted Answer

Because it accounts for the extra variability expected with small sample sizes.

Question 13

If you calculate a sample size $$n$$ that is not a whole number, what should you do?

Accepted Answer

Round $$n up to $$the next larger whole number.

Question 14

Which of the following is NOT a valid method for estimating the population standard deviation $$\sigma$$ when it is unknown?

Accepted Answer

Using the sample mean $$\bar{x} as an $$estimate for $$\sigma.$$

Question 15

A 95% confidence interval for the mean birth weight of girls is calculated as $$29.2 \leq \mu \leq 32.5 hg. $$What does this interval mean?

Accepted Answer

We are 95% confident that the interval contains the true mean birth weight of all girls.

Question 16

Which of the following best explains why the $$t$$ distribution approaches the standard normal distribution as $$n$$ increases?

Accepted Answer

Because the estimate of the population standard deviation becomes more accurate as sample size increases.

Estimating a Population Mean and Determining Sample Size: Confidence Intervals and the Student t Distribution

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