Confidence Intervals: Definition, Formula, and Interpretation

Question 1

A sample of 50 women has a mean height of 165 centimeters (cm). The standard deviation of the population is 5 cm. Construct a 95% confidence interval for the population mean height.

Accepted Answer

163.73 cm to 166.27 cm

Answer

164 cm to 166 cm

Answer

162.5 cm to 167.5 cm

Answer

162 cm to 168 cm

Question 2

The width of a confidence interval can be reduced by:

Accepted Answer

Increasing the sample size.

Answer

Ignoring outliers.

Answer

Decreasing the significance level.

Answer

Using a larger standard deviation.

Question 3

A 99% confidence interval is wider than a 95% confidence interval for the same population parameter and sample size. This is because:

Accepted Answer

The higher the confidence level, the wider the interval.

Answer

The larger the sample size, the wider the interval.

Answer

The lower the confidence level, the wider the interval.

Question 4

A confidence interval for a population mean is:

Accepted Answer

A range of values that is expected to include the true population mean.

Answer

A test statistic that determines if the sample mean is statistically significant.

Answer

A single value that is an estimate of the population mean.

Question 5

The confidence level of a confidence interval refers to the:

Accepted Answer

Probability that the true population parameter falls within the interval.

Answer

Probability that the sample mean falls within the interval.

Answer

Percentage of samples that will produce the same interval.

Question 6

In general, how does sample size impact the width of a confidence interval?

Accepted Answer

Larger sample sizes result in narrower confidence intervals.

Answer

Sample size has no impact on confidence interval width.

Answer

Larger sample sizes result in wider confidence intervals.

Question 7

Which of the following is a necessary condition for using the normal distribution to construct a confidence interval?

Accepted Answer

The sample size should be large, or the population distribution should be approximately normal.

Answer

The population distribution must be known exactly.

Answer

The sample size must be very large.

Question 8

What is the correct interpretation of a confidence interval?

Accepted Answer

We can be confident that the true population parameter falls within the interval with the specified level of confidence.

Answer

The interval represents the range of all possible sample means.

Answer

The population parameter is guaranteed to be exactly the midpoint of the interval.

Question 9

When comparing two confidence intervals, which one is more precise?

Accepted Answer

The interval with a smaller width.

Answer

Both intervals are equally precise.

Answer

Precision cannot be determined from the width alone.

Answer

The interval with a larger width.

Question 10

Which factors influence the width of a confidence interval?

Accepted Answer

Sample size, confidence level, and population variability

Answer

Population mean only

Answer

Confidence level only

Question 11

A confidence interval for a population proportion is calculated to be (0.4, 0.6). What does this interval imply?

Accepted Answer

With 95% confidence, the true population proportion is between 0.4 and 0.6.

Answer

The sample proportion is between 0.4 and 0.6.

Answer

The true population proportion is exactly 0.5.

Question 12

Which formula is used to calculate a confidence interval for the population mean?

Accepted Answer

x̄ ± z * (σ / √n)

Answer

t * s² / n ± z * √(t² * s² / n)

Answer

p ± z * √(p * q / n)

Answer

(s² / n) ± z * √(s²)

Question 13

What is the z-score for a 95% confidence level?

Accepted Answer

1.96

Answer

2.576

Answer

1.645

Question 14

What does a confidence interval represent?

Accepted Answer

A range of values within which the true population parameter is likely to lie

Answer

The exact value of the population parameter

Answer

A measure of how confident we are in our estimate

Question 15

How does a confidence interval differ from a hypothesis test?

Accepted Answer

A confidence interval estimates a parameter's value, while a hypothesis test determines a significant difference between groups

Answer

A confidence interval is more precise than a hypothesis test

Question 16

Which is an example of a biased sample?

Accepted Answer

A survey that includes only respondents who are willing to participate

Answer

A stratified sample

Answer

A random sample of the population

Question 17

How can the accuracy of a confidence interval be improved?

Accepted Answer

Increase the sample size

Answer

Decrease the confidence level

Answer

Increase the population variance

Answer

Decrease the sample size

Question 18

What is the level of confidence associated with a 90% confidence interval?

Accepted Answer

0.90

Answer

0.95

Answer

1.00

Answer

0.99

Question 19

A confidence interval for a population proportion has a lower bound of 0.10 and an upper bound of 0.20. What is the margin of error?

Accepted Answer

0.05

Answer

0.20

Answer

0.15

Answer

0.10

Question 20

Which of the following is a limitation of confidence intervals?

Accepted Answer

They are sensitive to outliers in the data.

Answer

They provide a range of plausible values for the population parameter.

Answer

They are easy to calculate.

Answer

They can be used to make predictions about future events.

Question 21

A confidence interval is considered to be statistically significant if:

Accepted Answer

It does not contain the hypothesized value of the population parameter.

Answer

It is wide.

Answer

It is narrow.

Answer

It is calculated using a large sample size.

Question 22

Which of the following is the best way to interpret a confidence interval?

Accepted Answer

We are 95% confident that the true population parameter is within the interval.

Answer

The interval represents the range of all possible values for the population parameter.

Answer

The true population parameter is guaranteed to be within the interval.