Question 1

Which of the following conditions must be met to use the t-distribution for constructing a confidence interval for a population mean?

Accepted Answer

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

Question 2

Given a sample mean $$\bar{x} = 50$$, sample standard deviation $$s = 8$$, sample size $$n = 25$$, and a 95% confidence level, what is the correct formula for the margin of error $$E$$ when the population standard deviation is unknown?

Accepted Answer

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

Question 3

A researcher wants to estimate the mean IQ of college professors. The IQ scores are normally distributed with a standard deviation of 15. What is the minimum sample size required to estimate the mean within 5 points with 95% confidence?

Accepted Answer

$$n = \left( \frac{2.575 \cdot 15}{5} \$$right)^2$$

Question 4

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

Accepted Answer

As the sample size increases, the t-distribution approaches the normal distribution.

Question 5

Suppose you have a sample of size $$n = 10$$ with a sample mean $$\bar{x} = 20$$ and sample standard deviation $$s = 4. $$What are the degrees of freedom for the t-distribution used to construct a confidence interval for the mean?

Accepted Answer

$$df = 9$$

Question 6

A 95% confidence interval for the mean download speed at an airport is calculated as (13.05, 22.15) Mbps. What is the point estimate of the population mean?

Accepted Answer

$$\mu = 17.60$$ Mbps

Question 7

If the confidence interval for the mean heart rate of males is (67.8, 71.4) bpm and for females is (72.0, 76.1) bpm, what can you infer about the difference in mean heart rates?

Accepted Answer

The mean heart rate for females is likely higher than for males.

Question 8

Which formula is used to determine the sample size $$n$$ required to estimate a population mean with a specified margin of error $$E$$ when the population standard deviation $$\sigma is $$known?

Accepted Answer

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

Question 9

When constructing a confidence interval for a population mean, which of the following is NOT a valid method for estimating the population standard deviation $$\sigma if it is $$unknown?

Accepted Answer

Assuming $$\sigma = 0 if $$the sample mean is large.

Question 10

A sample of 50 data points has a mean of 17.6 and a standard deviation of 1.6. What is the 95% confidence interval for the population mean? (Use $$t_{0.025, 49} \approx 2.0096)$$

Accepted Answer

$$(17.09, 18.11)$$

Question 11

Which of the following best describes the margin of error $$E in a $$confidence interval for a population mean?

Accepted Answer

It is the maximum likely difference between the sample mean and the true population mean at a given confidence level.

Question 12

If a sample of size $$n = 100 is $$drawn from a population with unknown mean and known standard deviation $$\sigma = 7$$, and the sample mean is $$\bar{x} = 205$$, what is the 95% confidence interval for the population mean? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$(203.62, 206.38)$$

Question 13

Why is the t-distribution used instead of the normal distribution when constructing a confidence interval for the mean with small samples?

Accepted Answer

Because the t-distribution accounts for the additional uncertainty in estimating the population standard deviation from a small sample.

Question 14

A biologist wants to estimate the mean number of DNA samples in a population. She collects a simple random sample of $$n = 6$$ with a sample mean of $$\bar{x} = 18.34$$ and sample standard deviation $$s = 1.0748. $$What is the correct formula for the 95% confidence interval for the mean?

Accepted Answer

$$\bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$$

Question 15

Which of the following is the correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

We are 95% confident that the interval contains the true population mean.

Question 16

If the sample size increases while keeping the confidence level and sample standard deviation constant, what happens to the margin of error $$E$$?

Accepted Answer

It decreases.

Question 17

Which of the following scenarios requires the use of the t-distribution rather than the normal distribution for constructing a confidence interval for the mean?

Accepted Answer

Sample size is 25 and population standard deviation is unknown.

Question 18

A sample of $$n = 36$$ has a mean of $$\bar{x} = 80$$ and a sample standard deviation $$s = 12. $$What is the 99% confidence interval for the population mean? (Use $$t_{0.005, 35} \approx 2.724)$$

Accepted Answer

$$(74.56, 85.44)$$

Question 19

Which of the following is NOT a requirement for constructing a confidence interval for a population mean using the t-distribution?

Accepted Answer

The population standard deviation must be known.

Estimating a Population Mean and Confidence Intervals

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