Question 1

The standard deviation of the sampling distribution of the sample mean is also called the:

Accepted Answer

The standard deviation of the sampling distribution of the sample mean is known as the standard error of the mean. This is an important concept in statistics because it tells us how much the sample mean is likely to vary from the population mean.

Question 2

Given an infinite population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations, taken at random from this population, exceeds 78 is:

Accepted Answer

We use the central limit theorem, which states that if the sample size is sufficiently large, the sampling distribution of the mean is approximately normal, regardless of the underlying distribution of the population.

First, we need to standardize the sample mean:

z = (x̄ - μ) / (σ/√n)
where x̄ = 78, μ = 75, σ = 12, and n = 36

z = (78 - 75) / (12/√36)
z = 1.5

Using a standard normal distribution table or calculator, we find that the probability of a z-score greater than 1.5 is approximately 0.0668.

Therefore, the answer is B) 0.0668.

Question 3

An infinite population has a mean of 33 and a standard deviation of 6. A sample of 100 observations is to be taken at random from this population. The probability that the sample mean will be between 34.5 and 36.1 is:

Accepted Answer

D

Question 4

A normally distributed population with 200 elements has a mean of 60 and a standard deviation of 10. The probability that the mean of a sample of 25 elements taken from this population will be smaller than 56 is:

Accepted Answer

The probability that the mean of a sample of 25 elements taken from this population will be smaller than 56 is 0.0166. This is because the mean of the sample will be normally distributed with a mean of 60 and a standard deviation of 10/sqrt(25) = 2. The probability that the mean of the sample will be smaller than 56 is the area under the normal distribution curve to the left of 56, which is 0.0166.

Question 5

A manufacturing company is concerned about the number of defective items produced by their assembly line. In the past they have had 5% of their products produced defectively. They take a random sample of 35 products. What is the probability that more than 5 products in the sample are defective?

Accepted Answer

This problem can be solved using the binomial distribution formula:
P(x > 5) = 1 - P(x ≤ 5)
P(x ≤ 5) = Σ (35 choose x) * 0.05^x * 0.95^(35-x) from x=0 to 5
P(x ≤ 5) = 0.4731
P(x > 5) = 1 - 0.4731
P(x > 5) = 0.5269 or 0.59% (rounded to two decimal places)
Therefore, the best choice is B.

Question 6

Which of the following best describes the Central Limit Theorem?

Accepted Answer

The Central Limit Theorem states that for large enough sample sizes ($n \geq 30$), the sampling distribution of the sample mean will be approximately normal, regardless of the underlying population distribution. Therefore, option A is the correct choice. Option B is incorrect because it makes a specific claim about the sample mean, which may or may not be true depending on the context. Option C is also incorrect because the Central Limit Theorem does not require the underlying population to be normal.

Question 7

It is know that 40% of voters in a certain electorate are in favour of a particular candidate. If a sample of size 30 is taken, what is the probability that less than 35% are in favour of this political candidate?

Accepted Answer

The probability can be calculated using the binomial distribution or normal approximation. The probability of less than 35% (10.5 voters) in favor is approximately 28.77%.

Question 8

If all possible samples of size n are drawn from a population, the probability distribution of the sample mean $\bar { X }$ is called the:

Accepted Answer

The probability distribution of the sample mean $\bar { X }$ is called the sampling distribution of $\bar { X }$, which is the distribution of all possible sample means of size n taken from the same population. It is important because it allows us to make statistical inferences about the population mean based on the sample mean. The standard error of $\bar { X }$ is the standard deviation of the sampling distribution of $\bar { X }$ and the expected value of $\bar { X }$ is equal to the population mean. The normal distribution is a specific type of distribution and not necessarily relevant to the sampling distribution of $\bar { X }$.

Question 9

If all possible samples of size n are drawn from an infinite population with mean$\mu$and standard deviation $\sigma$, then the standard error of the sample mean is inversely proportional to:

Accepted Answer

The standard error of the sample mean is given by σ/√n, where σ is the standard deviation of the population and n is the sample size. Therefore, the standard error is inversely proportional to the square root of n (i.e. 1/√n), which is represented by option D.

Question 10

A population that consists of 250 items has a mean of 37 and a standard deviation of 13. A sample of size 5 is taken at random from this population. The standard error of the sample mean equals (up to three decimal places):

Accepted Answer

The answer of A population that consists of 250 items...

Question 11

The expected value of the sampling distribution of the sample mean $\bar { X }$ equals the population mean$\mu$:

Accepted Answer

The expected value of the sampling distribution of the sample mean equals the population mean for all populations, regardless of their distribution or size. This is a fundamental concept in statistics known as the law of large numbers.

Question 12

As a general rule in computing the standard error of the sample mean, the finite population correction factor is used only if the:

Accepted Answer

The finite population correction factor is applied when the sample size is more than 5% of the population size, which is close to the condition described in option C.

Question 13

Consider an infinite population with a mean of 160 and a standard deviation of 25. A random sample of size 64 is taken from this population. The standard deviation of the sample mean equals:

Accepted Answer

The answer of Consider an infinite population with a mean...

Question 14

Which of the following best describes the sampling distribution of the sample proportion?

Accepted Answer

The sampling distribution of the sample proportion is approximately normally distributed provided that np≥ 5 and nq≥ 5. This is known as the normal approximation to the binomial distribution. If these conditions are met, the sampling distribution of the sample proportion will be bell-shaped and symmetric. If the conditions are not met, the sampling distribution may be skewed. Therefore, option B is the best choice as it covers both scenarios where the sampling distribution is approximately normally distributed and when it is skewed (if the conditions are not met). Option A is not correct as the sampling distribution is not always normally distributed. Option D is not correct as one of the options is correct.

Question 15

A sample of size 50 is to be taken from an infinite population whose mean and standard deviation are 52 and 20, respectively. The probability that the sample mean will be larger than 49 is:

Accepted Answer

Since the sample size (n=50) is large, we can use the Central Limit Theorem to approximate the sample mean as a normal distribution with mean 52 and standard deviation 20/sqrt(50)=2.8284.

Then, we can standardize the distribution using z-score:

z = (49-52)/2.8284 = -1.0607

The probability of the sample mean being larger than 49 is equivalent to the probability of z being larger than -1.0607. From standard normal distribution table, we can find that:

P(z > -1.0607) = 0.8554

Therefore, the answer is C.

Question 16

Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 45 and 9, respectively. The mean and standard error of the sampling distribution of the sample mean are:

Accepted Answer

The mean of the sampling distribution of the sample mean is equal to the population mean, which is 45. The standard error of the sampling distribution is calculated as the population standard deviation divided by the square root of the sample size, which is 9 / sqrt(81) = 9 / 9 = 1.

Question 17

A sample of size n is selected at random from an infinite population. As n increases, which of the following statements is true?

Accepted Answer

As the sample size increases, the standard error of the sample mean decreases because the standard error is inversely proportional to the square root of the sample size. The population standard deviation does not depend on the sample size, so option A and C can be ruled out. Option D is incorrect because the standard error of the sample mean decreases with an increase in sample size.

Question 18

The finite population correction factor should not be used when:

Accepted Answer

The finite population correction factor is used when sampling from a finite population to adjust for the decrease in variability that occurs when the sample size approaches the population size. However, if we are sampling from an infinite population, there is no need to adjust for this decrease in variability, and therefore the finite population correction factor should not be used. Option B and C are incorrect because they don't cover all scenarios when the correction factor should not be used.

Question 19

Random samples of size 64 are taken from an infinite population whose mean is 160 and standard deviation is 32. The mean and standard error of the sample mean, respectively, are:

Accepted Answer

The mean of the sample mean is the same as the population mean which is 160. 
The standard error of the sample mean can be calculated using the formula:

SE = SD / sqrt(n)

where SD is the population standard deviation and n is the sample size.

Thus, SE = 32 / sqrt(64) = 4.

Therefore, the correct answer is C, with mean 160 and standard error 4.

Question 20

If a random sample of size n is drawn from a normal population, then the sampling distribution of the sample mean $\bar { X }$ will be:

Accepted Answer

The sampling distribution of the sample mean $\bar{X}$ is normal for all values of n if the population is normal, due to the properties of the normal distribution.

Quiz 10: Sampling Distributions

The standard deviation of the sampling distribution of the sample mean is also called the:

Given an infinite population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations, taken at random from this population, exceeds 78 is:

An infinite population has a mean of 33 and a standard deviation of 6. A sample of 100 observations is to be taken at random from this population. The probability that the sample mean will be between 34.5 and 36.1 is:

A normally distributed population with 200 elements has a mean of 60 and a standard deviation of 10. The probability that the mean of a sample of 25 elements taken from this population will be smaller than 56 is:

A manufacturing company is concerned about the number of defective items produced by their assembly line. In the past they have had 5% of their products produced defectively. They take a random sample of 35 products. What is the probability that more than 5 products in the sample are defective?

Which of the following best describes the Central Limit Theorem?

It is know that 40% of voters in a certain electorate are in favour of a particular candidate. If a sample of size 30 is taken, what is the probability that less than 35% are in favour of this political candidate?

If all possible samples of size n are drawn from a population, the probability distribution of the sample mean $\bar { X }$ is called the:

If all possible samples of size n are drawn from an infinite population with mean $\mu$ and standard deviation $\sigma$ , then the standard error of the sample mean is inversely proportional to:

A population that consists of 250 items has a mean of 37 and a standard deviation of 13. A sample of size 5 is taken at random from this population. The standard error of the sample mean equals (up to three decimal places) :

The expected value of the sampling distribution of the sample mean $\bar { X }$ equals the population mean $\mu$ :

As a general rule in computing the standard error of the sample mean, the finite population correction factor is used only if the:

Consider an infinite population with a mean of 160 and a standard deviation of 25. A random sample of size 64 is taken from this population. The standard deviation of the sample mean equals:

Which of the following best describes the sampling distribution of the sample proportion?

A sample of size 50 is to be taken from an infinite population whose mean and standard deviation are 52 and 20, respectively. The probability that the sample mean will be larger than 49 is:

Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 45 and 9, respectively. The mean and standard error of the sampling distribution of the sample mean are:

A sample of size n is selected at random from an infinite population. As n increases, which of the following statements is true?

The finite population correction factor should not be used when:

Random samples of size 64 are taken from an infinite population whose mean is 160 and standard deviation is 32. The mean and standard error of the sample mean, respectively, are:

If a random sample of size n is drawn from a normal population, then the sampling distribution of the sample mean $\bar { X }$ will be:

Quiz 10: Sampling Distributions

The standard deviation of the sampling distribution of the sample mean is also called the:

Given an infinite population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations, taken at random from this population, exceeds 78 is:

An infinite population has a mean of 33 and a standard deviation of 6. A sample of 100 observations is to be taken at random from this population. The probability that the sample mean will be between 34.5 and 36.1 is:

A normally distributed population with 200 elements has a mean of 60 and a standard deviation of 10. The probability that the mean of a sample of 25 elements taken from this population will be smaller than 56 is:

A manufacturing company is concerned about the number of defective items produced by their assembly line. In the past they have had 5% of their products produced defectively. They take a random sample of 35 products. What is the probability that more than 5 products in the sample are defective?

Which of the following best describes the Central Limit Theorem?

It is know that 40% of voters in a certain electorate are in favour of a particular candidate. If a sample of size 30 is taken, what is the probability that less than 35% are in favour of this political candidate?

If all possible samples of size n are drawn from a population, the probability distribution of the sample mean Xˉ\bar { X }Xˉ is called the:

If all possible samples of size n are drawn from an infinite population with mean μ\muμ and standard deviation σ\sigmaσ , then the standard error of the sample mean is inversely proportional to:

A population that consists of 250 items has a mean of 37 and a standard deviation of 13. A sample of size 5 is taken at random from this population. The standard error of the sample mean equals (up to three decimal places) :

The expected value of the sampling distribution of the sample mean Xˉ\bar { X }Xˉ equals the population mean μ\muμ :

As a general rule in computing the standard error of the sample mean, the finite population correction factor is used only if the:

Consider an infinite population with a mean of 160 and a standard deviation of 25. A random sample of size 64 is taken from this population. The standard deviation of the sample mean equals:

Which of the following best describes the sampling distribution of the sample proportion?

A sample of size 50 is to be taken from an infinite population whose mean and standard deviation are 52 and 20, respectively. The probability that the sample mean will be larger than 49 is:

Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 45 and 9, respectively. The mean and standard error of the sampling distribution of the sample mean are:

A sample of size n is selected at random from an infinite population. As n increases, which of the following statements is true?

The finite population correction factor should not be used when:

Random samples of size 64 are taken from an infinite population whose mean is 160 and standard deviation is 32. The mean and standard error of the sample mean, respectively, are:

If a random sample of size n is drawn from a normal population, then the sampling distribution of the sample mean Xˉ\bar { X }Xˉ will be:

If all possible samples of size n are drawn from a population, the probability distribution of the sample mean $\bar { X }$ is called the:

If all possible samples of size n are drawn from an infinite population with mean $\mu$ and standard deviation $\sigma$ , then the standard error of the sample mean is inversely proportional to:

The expected value of the sampling distribution of the sample mean $\bar { X }$ equals the population mean $\mu$ :

If a random sample of size n is drawn from a normal population, then the sampling distribution of the sample mean $\bar { X }$ will be: