Question 1

Which of the following best describes a sampling distribution?

Accepted Answer

A distribution of all possible values of a sample statistic for a given sample size selected from a population.

Question 2

Suppose the population mean is $\mu = 50$ and the population standard deviation is $\sigma = 10$. If you take a random sample of size $n = 25$, what is the standard error of the mean?

Accepted Answer

$2$

Question 3

Which of the following statements about the Central Limit Theorem (CLT) is correct?

Accepted Answer

The CLT states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Question 4

A restaurant has a mean wait time of $\mu = 8$ minutes and a standard deviation of $\sigma = 3$ minutes. If a random sample of $n = 36$ customers is taken, what is the probability that the sample mean wait time is between $7.5$ and $8.5$ minutes? (Assume normality.)

Accepted Answer

Approximately $0.8664$

Question 5

If the population is not normally distributed, under what condition can we use the normal distribution to approximate the sampling distribution of the sample mean?

Accepted Answer

When the sample size is large, typically $n > 30$.

Question 6

Given a population proportion $\pi = 0.4$ and a sample size $n = 200$, what is the standard error of the sample proportion?

Accepted Answer

$0.0346$

Question 7

Which formula is used to calculate the $Z$-value for the sampling distribution of the sample mean?

Accepted Answer

$Z = \frac{\overline{X} - \mu}{\sigma / \sqrt{n}}$

Question 8

A company wants to estimate the proportion of customers who prefer online shopping. If the true proportion is $\pi = 0.6$ and a sample of $n = 100$ is taken, what is the probability that the sample proportion is between $0.55$ and $0.65$? (Use the normal approximation.)

Accepted Answer

Approximately $0.7287$

Question 9

Which of the following is NOT a property of the sampling distribution of the sample mean?

Accepted Answer

The standard deviation of the sampling distribution increases as sample size increases.

Question 10

If a population has a mean $\mu = 100$ and standard deviation $\sigma = 20$, what is the probability that a sample mean from a sample of $n = 64$ falls within $5$ units of the population mean? (Assume normality.)

Accepted Answer

Approximately $0.7888$

Question 11

Which of the following scenarios best illustrates the concept of an unbiased estimator?

Accepted Answer

The mean of all possible sample means equals the population mean.

Question 12

A sample of size $n = 25$ is drawn from a population with $\mu = 368$ and $\sigma = 15$. What is the 95% confidence interval for the sample mean?

Accepted Answer

$[362.12, 373.88]$

Question 13

Which of the following is the correct formula for the standard error of the sample mean?

Accepted Answer

$\frac{\sigma}{\sqrt{n}}$

Question 14

If a population is highly skewed, what is the minimum sample size generally recommended for the sampling distribution of the mean to be approximately normal?

Accepted Answer

$n > 30$

Question 15

A sample of $n = 100$ is taken from a population with $\pi = 0.2$. What is the probability that the sample proportion is greater than $0.25$? (Use the normal approximation.)

Accepted Answer

Approximately $0.1056$

Question 16

Which of the following is a correct interpretation of the Central Limit Theorem in business applications?

Accepted Answer

It allows us to use normal probability models for sample means, even when the population is not normal, provided the sample size is large.

Question 17

A population has $N = 4$ with values $18, 20, 22, 24$. If all possible samples of size $n = 2$ are taken with replacement, what is the mean of the sampling distribution of the sample mean?

Accepted Answer

$21$

Question 18

Which of the following is a requirement for using the normal approximation for the sampling distribution of the sample proportion?

Accepted Answer

$n\pi > 5$ and $n(1-\pi) > 5$

Question 19

If the sample size increases, what happens to the standard error of the mean?

Accepted Answer

It decreases.

Question 20

A business analyst wants to know the probability that the sample mean sales from a sample of $n = 49$ stores is at least $2$ units above the population mean, given $\sigma = 14$. What is the $Z$-value for this scenario?

Accepted Answer

$1$

Sampling Distributions: Concepts and Applications in Business Statistics

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