Q: Which property distinguishes the Student's $t$-distribution from the standard normal distribution?

The $t$-distribution has fatter tails, especially for small sample sizes

Q: A sample of $n=20$ long-distance phone calls has a sample mean duration of $12.5$ minutes and sample standard deviation $s=5$ minutes. What is the 95% confidence interval for the mean duration time? (Use $t_{0.025,19}=2.093$)

$(12.5 - 2.093 \frac{5}{\sqrt{20}},\ 12.5 + 2.093 \frac{5}{\sqrt{20}}) = (10.16, 14.84)$

Q: Which of the following is NOT a requirement for constructing a confidence interval for a population proportion?

The sample mean must be greater than the population mean

Q: For a small sample ($n < 30$) with unknown population standard deviation, which formula should be used to construct a $(1-\alpha)100\%$ confidence interval for the mean $\mu$?

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

Q: Why does the Student's $t$-distribution have fatter tails than the standard normal distribution?

Because the sample standard deviation $s$ is used to estimate $\sigma$, introducing more variability

Question 1

A researcher wants to estimate the proportion of college students who own a car. She surveys 200 students and finds that 60 own a car. What is the 95% confidence interval for the true proportion of college students who own a car? (Use $z_{\alpha/2} = 1.96$)

Accepted Answer

$0.30 \pm 1.96 \sqrt{\frac{0.30(1-0.30)}{200}} = (0.236, 0.364)$

Question 2

Which of the following will result in a narrower confidence interval for a population proportion, assuming all other factors are held constant?

Accepted Answer

Increasing the sample size $n$

Question 3

A survey of 800 teenagers found that 272 text while driving. What is the margin of error for a 95% confidence interval for the proportion who text while driving? (Use $z_{\alpha/2} = 1.96$)

Accepted Answer

$E = 1.96 \sqrt{\frac{0.34(1-0.34)}{800}} = 0.033$

Question 4

If the level of confidence is increased from 95% to 99% for a given sample, what happens to the margin of error and the width of the confidence interval?

Accepted Answer

Both the margin of error and the width of the confidence interval increase

Question 5

An economist wants to estimate the proportion of U.S. residents who carpool to work. If no prior estimate is available, what value should be used for $\hat{p}$ to determine the required sample size for a given margin of error and confidence level?

Accepted Answer

$\hat{p} = 0.5$

Question 6

Which formula gives the minimum sample size $n$ required to estimate a population proportion with margin of error $E$ and confidence level $(1-\alpha)100\%$, when no prior estimate of $\hat{p}$ is available?

Accepted Answer

$n = 0.25 \left( \frac{z_{\alpha/2}}{E} \right)^2$

Question 7

A random sample of size $n=100$ from a population with unknown mean $\mu$ and known standard deviation $\sigma=12$ yields a sample mean $\bar{x}=72.5$. What is the 95% confidence interval for $\mu$? (Use $z_{\alpha/2}=1.96$)

Accepted Answer

$(72.5 - 1.96 \frac{12}{\sqrt{100}},\ 72.5 + 1.96 \frac{12}{\sqrt{100}}) = (70.15, 74.85)$

Question 8

When constructing a confidence interval for the population mean $\mu$ with a small sample ($n < 30$) and unknown population standard deviation, which distribution should be used?

Accepted Answer

Student's $t$-distribution

Question 9

Which property distinguishes the Student's $t$-distribution from the standard normal distribution?

Accepted Answer

The $t$-distribution has fatter tails, especially for small sample sizes

Question 10

A sample of $n=20$ long-distance phone calls has a sample mean duration of $12.5$ minutes and sample standard deviation $s=5$ minutes. What is the 95% confidence interval for the mean duration time? (Use $t_{0.025,19}=2.093$)

Accepted Answer

$(12.5 - 2.093 \frac{5}{\sqrt{20}},\ 12.5 + 2.093 \frac{5}{\sqrt{20}}) = (10.16, 14.84)$

Question 11

Which of the following is NOT a requirement for constructing a confidence interval for a population proportion?

Accepted Answer

The sample mean must be greater than the population mean

Question 12

Which of the following factors does NOT increase the margin of error in a confidence interval for a population mean?

Accepted Answer

Increasing the sample size $n$

Question 13

For a small sample ($n < 30$) with unknown population standard deviation, which formula should be used to construct a $(1-\alpha)100\%$ confidence interval for the mean $\mu$?

Accepted Answer

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

Question 14

Why does the Student's $t$-distribution have fatter tails than the standard normal distribution?

Accepted Answer

Because the sample standard deviation $s$ is used to estimate $\sigma$, introducing more variability

Question 15

Which statement best describes the interpretation of a 95% confidence interval for a population mean $\mu$?

Accepted Answer

If we construct 100 such intervals from repeated samples, about 95 of them will contain the true mean $\mu$

Question 16

A sample of $n=50$ is drawn from a population with unknown mean and unknown standard deviation. Which distribution should be used to construct a confidence interval for the mean?

Accepted Answer

Standard normal distribution (using $z$-values)

Question 17

Which of the following is a point estimate for the population mean $\mu$?

Accepted Answer

$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$

Question 18

When is it appropriate to use the Student's $t$-distribution for constructing a confidence interval for the mean?

Accepted Answer

When the sample size is small ($n < 30$) and the population standard deviation is unknown

Lecture 25: Confidence Intervals – Proportions, Means, and the t-Distribution

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