Sampling Distributions and CLT Quiz

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2. Two sampling distributions are being compared: one from samples of size 50 and another from samples of size 200, both from the same population. Which distribution has a smaller standard error?

The distribution from n = 50

The distribution from n = 200

Both have the same standard error

Cannot be determined without knowing σ

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4. The shape of a sampling distribution of the sample mean depends primarily on the sample size and the shape of the original population. For which condition is normality guaranteed even for small samples?

When the population is normally distributed

When the sample is random

When the standard deviation is known

When n ≥ 30

5. A researcher draws samples of size 36 from a population with mean 75 and standard deviation 12. What is the standard error of the sample mean?

0.33

2

3

12

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7. If the population distribution is skewed, the sampling distribution of the sample mean will also be skewed regardless of sample size.

True

False

8. Which of the following best describes why the Central Limit Theorem is crucial in statistics?

It guarantees that all samples are representative of the population

It allows us to make inferences about population parameters using sample data

It eliminates the need for random sampling

It proves that the sample mean always equals the population mean

9. The margin of error in confidence intervals is directly proportional to the standard error of the sample mean.

True

False

10. According to the Central Limit Theorem, for large sample sizes, the sampling distribution of the sample mean is approximately normal with mean μ and standard deviation σ/√n. Which parameter represents the variability of individual observations?

μ

σ

σ/√n

√n

11. A sampling distribution represents the distribution of a statistic calculated from repeated samples of size n from a population. What is the mean of the sampling distribution of the sample mean equal to?

The population mean μ

The sample mean x̄

The standard deviation σ

The sample size n

12. The sampling distribution of the sample proportion p̂ is approximately normal when np ≥ 10 and n(1-p) ≥ 10. What do these conditions ensure?

The sample size is sufficiently large

Both success and failure counts are adequate

The population proportion is known

The sample is representative of the population

13. If you increase the sample size from n = 50 to n = 200, how does the standard error of the sample mean change?

It increases by a factor of 4

It decreases by a factor of 2

It remains unchanged

It increases by a factor of 2

14. The Central Limit Theorem requires that the population distribution must be normal for the theorem to apply.

True

False

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16. An unbiased estimator is one whose expected value equals the parameter being estimated. Is the sample mean an unbiased estimator of the population mean?

True

False

17. Which of the following statements about sampling distributions is correct?

The spread of a sampling distribution increases with sample size

The mean of the sampling distribution is typically different from the population mean

The sampling distribution of the sample mean is approximately normal for large n

The standard error equals the population standard deviation

18. If a population has mean μ = 100 and standard deviation σ = 15, what is the standard error of the sample mean for samples of size n = 25?

0.6

3

6

15

19. The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as n increases, regardless of the shape of the population distribution. This is true for which condition?

Only if the population is normally distributed

Only if n is very large (typically n ≥ 30)

Only if the population standard deviation is known

Only if the sample is randomly selected

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DataAI Sampling Distributions and CLT Quiz

1. A population has mean 50 and standard deviation 8. For samples of size 16, the sampling distribution of x̄ has standard error ____.

2.

What first name or nickname would you like us to use?

2. Two sampling distributions are being compared: one from samples of size 50 and another from samples of size 200, both from the same population. Which distribution has a smaller standard error?

3. As sample size increases, the sampling distribution of the sample mean becomes more concentrated around the population mean. This property is a consequence of the ____.

4. The shape of a sampling distribution of the sample mean depends primarily on the sample size and the shape of the original population. For which condition is normality guaranteed even for small samples?

5. A researcher draws samples of size 36 from a population with mean 75 and standard deviation 12. What is the standard error of the sample mean?

6. The standard error of a statistic measures the variability of that statistic across repeated samples. For the sample mean, this variability decreases as ____.

7. If the population distribution is skewed, the sampling distribution of the sample mean will also be skewed regardless of sample size.

8. Which of the following best describes why the Central Limit Theorem is crucial in statistics?

9. The margin of error in confidence intervals is directly proportional to the standard error of the sample mean.

10. According to the Central Limit Theorem, for large sample sizes, the sampling distribution of the sample mean is approximately normal with mean μ and standard deviation σ/√n. Which parameter represents the variability of individual observations?

11. A sampling distribution represents the distribution of a statistic calculated from repeated samples of size n from a population. What is the mean of the sampling distribution of the sample mean equal to?

12. The sampling distribution of the sample proportion p̂ is approximately normal when np ≥ 10 and n(1-p) ≥ 10. What do these conditions ensure?

13. If you increase the sample size from n = 50 to n = 200, how does the standard error of the sample mean change?

14. The Central Limit Theorem requires that the population distribution must be normal for the theorem to apply.

15. For a normally distributed population with σ = 10, samples of size n = 100 are drawn. The standard error of the sample mean is ____.

16. An unbiased estimator is one whose expected value equals the parameter being estimated. Is the sample mean an unbiased estimator of the population mean?

17. Which of the following statements about sampling distributions is correct?

18. If a population has mean μ = 100 and standard deviation σ = 15, what is the standard error of the sample mean for samples of size n = 25?

19. The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as n increases, regardless of the shape of the population distribution. This is true for which condition?

20. The standard error of the sample mean is σ/√n. As sample size increases, the standard error ____.