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Interval Estimation

Description: This quiz will test your understanding of interval estimation, a fundamental concept in statistics used to estimate the range of possible values for a population parameter based on sample data.
Number of Questions: 15
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Tags: interval estimation confidence intervals hypothesis testing sampling
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What is the purpose of interval estimation?

  1. To determine the exact value of a population parameter

  2. To estimate the range of possible values for a population parameter

  3. To test the significance of a difference between two sample means

  4. To determine the probability of obtaining a sample mean

Show answer
Correct Option: B
Explanation:

Interval estimation aims to provide a range of values within which the true population parameter is likely to fall, rather than determining its exact value.

Which of the following is a commonly used method for constructing confidence intervals?

  1. Point estimation

  2. Hypothesis testing

  3. Bootstrapping

  4. Regression analysis

Show answer
Correct Option: C
Explanation:

Bootstrapping is a resampling technique used to estimate the distribution of a statistic by repeatedly drawing samples from the original data.

What is the interpretation of a 95% confidence interval?

  1. There is a 95% chance that the true population parameter is within the interval

  2. The probability of obtaining a sample mean within the interval is 95%

  3. The interval contains the true population parameter with a probability of 95%

  4. The sample mean is within the interval with a probability of 95%

Show answer
Correct Option: A
Explanation:

A 95% confidence interval means that if we were to repeatedly take samples from the population and construct confidence intervals, 95% of those intervals would contain the true population parameter.

Which of the following factors affects the width of a confidence interval?

  1. Sample size

  2. Level of confidence

  3. Population standard deviation

  4. All of the above

Show answer
Correct Option: D
Explanation:

The width of a confidence interval is influenced by the sample size, the level of confidence, and the population standard deviation.

What is the relationship between sample size and the width of a confidence interval?

  1. As sample size increases, the width of the confidence interval decreases

  2. As sample size increases, the width of the confidence interval increases

  3. Sample size has no effect on the width of the confidence interval

  4. The relationship between sample size and the width of the confidence interval depends on the level of confidence

Show answer
Correct Option: A
Explanation:

Generally, as the sample size increases, the width of the confidence interval decreases, making the interval more precise.

Which of the following is true about the level of confidence in interval estimation?

  1. A higher level of confidence results in a wider confidence interval

  2. A higher level of confidence results in a narrower confidence interval

  3. The level of confidence has no effect on the width of the confidence interval

  4. The relationship between the level of confidence and the width of the confidence interval depends on the sample size

Show answer
Correct Option: A
Explanation:

As the level of confidence increases, the width of the confidence interval also increases, making the interval less precise.

What is the effect of population standard deviation on the width of a confidence interval?

  1. As population standard deviation increases, the width of the confidence interval decreases

  2. As population standard deviation increases, the width of the confidence interval increases

  3. Population standard deviation has no effect on the width of the confidence interval

  4. The relationship between population standard deviation and the width of the confidence interval depends on the sample size

Show answer
Correct Option: B
Explanation:

A larger population standard deviation leads to a wider confidence interval, making the interval less precise.

What is the difference between a confidence interval and a hypothesis test?

  1. A confidence interval estimates the range of possible values for a population parameter, while a hypothesis test determines whether a particular value is plausible

  2. A confidence interval determines whether a particular value is plausible, while a hypothesis test estimates the range of possible values for a population parameter

  3. Both confidence intervals and hypothesis tests estimate the range of possible values for a population parameter

  4. Both confidence intervals and hypothesis tests determine whether a particular value is plausible

Show answer
Correct Option: A
Explanation:

Confidence intervals provide a range of plausible values for a population parameter, while hypothesis tests evaluate whether a specific claim about the population parameter is supported by the data.

Which of the following is a common type of confidence interval for a population mean?

  1. t-interval

  2. z-interval

  3. F-interval

  4. chi-square interval

Show answer
Correct Option: A
Explanation:

t-intervals are commonly used when the population standard deviation is unknown and the sample size is small (less than 30).

What is the formula for calculating the margin of error in a confidence interval?

  1. Margin of error = (Critical value) * (Standard error of the mean)

  2. Margin of error = (Critical value) * (Sample standard deviation)

  3. Margin of error = (Critical value) * (Population standard deviation)

  4. Margin of error = (Critical value) * (Sample mean)

Show answer
Correct Option: A
Explanation:

The margin of error is calculated by multiplying the critical value from the appropriate distribution (t-distribution or z-distribution) by the standard error of the mean.

What is the critical value in the context of confidence intervals?

  1. The value that divides the rejection region from the non-rejection region in a hypothesis test

  2. The value that corresponds to the desired level of confidence in a confidence interval

  3. The value that represents the center of the confidence interval

  4. The value that represents the width of the confidence interval

Show answer
Correct Option: B
Explanation:

The critical value is obtained from the appropriate distribution (t-distribution or z-distribution) based on the desired level of confidence.

What is the standard error of the mean?

  1. The standard deviation of the sample mean

  2. The standard deviation of the population mean

  3. The average of the sample means

  4. The difference between the sample mean and the population mean

Show answer
Correct Option: A
Explanation:

The standard error of the mean is a measure of the variability of the sample mean and is calculated by dividing the population standard deviation by the square root of the sample size.

What is the relationship between the sample size and the standard error of the mean?

  1. As sample size increases, the standard error of the mean increases

  2. As sample size increases, the standard error of the mean decreases

  3. Sample size has no effect on the standard error of the mean

  4. The relationship between sample size and the standard error of the mean depends on the population standard deviation

Show answer
Correct Option: B
Explanation:

As the sample size increases, the standard error of the mean decreases, making the sample mean more precise.

Which of the following is true about the interpretation of a confidence interval?

  1. If a confidence interval contains the hypothesized value, we can conclude that the hypothesis is true

  2. If a confidence interval does not contain the hypothesized value, we can conclude that the hypothesis is false

  3. A confidence interval provides a range of plausible values for the population parameter

  4. A confidence interval tells us the exact value of the population parameter

Show answer
Correct Option: C
Explanation:

Confidence intervals provide a range of plausible values for the population parameter, but they do not confirm or reject hypotheses.

What is the purpose of hypothesis testing in relation to interval estimation?

  1. To determine the exact value of a population parameter

  2. To estimate the range of possible values for a population parameter

  3. To evaluate whether a particular claim about a population parameter is supported by the data

  4. To determine the probability of obtaining a sample mean

Show answer
Correct Option: C
Explanation:

Hypothesis testing is used to determine whether a specific claim about a population parameter is supported by the evidence provided by the sample data.

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