Question 1

A sample is selected from a normal population with μ = 50 and σ = 12.Which of the following samples would be considered extreme and unrepresentative for this population?
A)M = 53 and n = 16
B)M = 53 and n = 4
C)M = 56 and n = 16
D)M = 56 and n = 4

Accepted Answer

The answer of A sample is selected from a normal...

Question 2

Two samples probably will have different means even if they are both the same size and they are both selected from the same population.

Accepted Answer

This is because samples are selected randomly and by chance, so it is possible for two different samples to have slightly different characteristics even if they are both from the same population. These differences can result in different sample means.

Question 3

If samples are selected from a normal population,the distribution of sample means will also be normal.

Accepted Answer

This is true due to the Central Limit Theorem, which states that as sample size increases, the distribution of sample means will approach a normal distribution, regardless of the underlying population distribution.

Question 4

A sample is selected from a population with a mean of µ = 40.If the sample mean is M = 45,which of the following combinations would make the sample mean a good,representative value for the population?
A)A small sample and a small population standard deviation
B)A small sample and a large population standard deviation
C)A large sample and a small population standard deviation
D)A large sample and a large population standard deviation

Accepted Answer

The answer of A sample is selected from a population...

Question 5

Samples of n = 9 scores are selected from a population.If the distribution of sample means has an expected value of 40,then the population has a mean of μ = 40.

Accepted Answer

The fact that the distribution of sample means has an expected value of 40 implies that the mean of the population is also 40. This is because the expected value of a sample mean is equal to the population mean when the sample is drawn from a normally distributed population. Therefore, the answer is true.

Question 6

If the sample size is equal to the population variance (n = σ²),then the standard error is equal to 1.

Accepted Answer

The standard error is defined as the standard deviation of the sampling distribution of the sample mean. The standard deviation of the sampling distribution is equal to the population standard deviation divided by the square root of the sample size. Therefore, if the sample size is equal to the population variance, then the standard error is equal to 1.

Question 7

A sample is selected from a population with a mean of µ = 40.If the sample mean is M = 45,which of the following combinations would make the sample mean an extreme,unrepresentative value for the population?
A)A small sample and a small population standard deviation
B)A small sample and a large population standard deviation
C)A large sample and a small population standard deviation
D)A large sample and a large population standard deviation

Accepted Answer

The answer of A sample is selected from a population...

Question 8

A sample of n = 4 scores is selected from a normal population with a mean of µ = 50 and a standard deviation of σ = 20.What is the probability of obtaining a sample mean greater than M = 48?
A)p = 0.6554
B)p = 0.3446
C)p = 0.5793
D)p = 0.4207

Accepted Answer

The answer of A sample of n = 4 scores...

Question 9

Which combination of factors will produce the largest value for the standard error?
A)A large sample and a large standard deviation
B)A small sample and a large standard deviation
C)A large sample and a small standard deviation
D)A small sample and a small standard deviation

Accepted Answer

Standard error is calculated as the standard deviation of the sample divided by the square root of the sample size. Thus, a smaller sample size will lead to a larger standard error. Additionally, a larger standard deviation means that the data points are more spread out from the mean, which can lead to a larger standard error as well. Therefore, option B (a small sample and a large standard deviation) will produce the largest value for the standard error.

Question 10

Samples of n = 4 scores are selected from a population.If the distribution of sample means has a standard error σ = 5 points,then the population from which the samples were obtained has a standard deviation of σ = 20.

Accepted Answer

The standard error of the mean (σM) is calculated as the population standard deviation (σ) divided by the square root of the sample size (n), so if σM = 5 and n = 4, then σ = 5 * sqrt(4) = 10, not 20.

Question 11

A random sample of n = 9 scores is selected from a normal distribution with μ = 80 and σ = 12.What is the probability that the sample mean will be between 76 and 84?
A)0.9974
B)0.6826
C)0.3830
D)0.2586

Accepted Answer

We can use the central limit theorem to approximate the distribution of sample means as normal with mean μ and standard deviation σ/sqrt(n). 
Thus, for this problem, we have:
mean = μ = 80
standard deviation = σ/sqrt(n) = 12/sqrt(9) = 4
Now, we need to find the probability that the sample mean, x̄, is between 76 and 84. 
To do this, we standardize the values:
z1 = (76-80)/4 = -1
z2 = (84-80)/4 = 1
Then, we use a normal distribution table or calculator to find the probability that a standard normal variable is between -1 and 1 (i.e., the area under the curve between z1 and z2). This probability is approximately 0.6826. Therefore, the answer is B.

Question 12

A random sample of n = 16 scores is selected from a normal distribution with μ = 500 and σ = 200.For this sample,which of the following is true?
A)p(402 < M < 598) = 0.95
B)p(425 < M < 575) = 0.95
C)p(450 < M < 550) = 0.95
D)p(490 < M < 510) = 0.95

Accepted Answer

The correct choice is based on the calculation of the probability that the sample mean (M) falls within a certain range, using the standard error of the mean (σM = σ / √n) and the properties of the normal distribution. Given σ = 200 and n = 16, the standard error σM = 200 / √16 = 50. The z-score for a 95% confidence interval around the mean (μ = 500) in a standard normal distribution is approximately ±1.96. Thus, the range for M is μ ± 1.96*σM = 500 ± 1.96*50 = 500 ± 98, which gives the range from 402 to 598, making option A correct.

Question 13

The mean for the distribution of sample means is always equal to the mean for the population from which the samples are obtained.

Accepted Answer

This is known as the law of large numbers, which states that as the sample size increases, the sample mean approaches the population mean. Therefore, the mean for the distribution of sample means will always be equal to the mean for the population.

Question 14

In order for the distribution of sample means to be normal,it must be based on samples of at least n = 30 scores.

Accepted Answer

The distribution of sample means will be approximately normal if the sample size is large enough, typically n ≥ 30, but it can also be normal for smaller sample sizes if the population distribution is normal.

Question 15

According to the central limit theorem,the standard error for a sample mean becomes smaller as the sample size increases.

Accepted Answer

The central limit theorem states that as the sample size increases, the distribution of the sample mean becomes more normal and the standard error of the mean decreases. Therefore, the statement is true.

Question 16

A sample of n = 25 scores is selected from a population with a mean of µ = 80 and a standard deviation of σ = 20.The standard error for the sample mean is 20.

Accepted Answer

The standard error for the sample mean is not 20, but rather determined by the formula SE = σ/√n. Plugging in the values given, we get SE = 20/√25 = 4.

Question 17

Samples of n = 16 are selected from a population.If the distribution of sample means has a standard error of 2 points,then the population has a standard deviation of σ = 8.

Accepted Answer

The standard error (SE) of the mean for a sample of size n is given by SE = σ/√n, where σ is the population standard deviation.

Here, we are given SE = 2 and n = 16.

So, 2 = 8/√16

Simplifying this equation, we get 2 = 8/4, which is true.

Therefore, the answer is A (True).

Question 18

A sample of n = 16 scores is selected from a population with μ = 100 and σ = 32.If the sample mean is M = 104,what is the z-score for this sample mean?
A)2.00
B)1.00
C)0.50
D)0.25

Accepted Answer

The z-score for a sample mean is calculated using the formula $z = \frac{M - \mu}{\sigma/\sqrt{n}}$, where $M$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size. Plugging in the given values: $z = \frac{104 - 100}{32/\sqrt{16}} = \frac{4}{8} = 0.50$.

Question 19

What happens to the expected value of M as sample size increases?
A)It increases.
B)It decreases.
C)It stays constant.
D)The expected value does not change in a predictable manner when sample size increases.

Accepted Answer

The expected value of a statistic like the mean (M) is a theoretical value based on the population from which samples are drawn. Increasing the sample size does not change this theoretical expectation; it only potentially reduces the variance around the expected value, making estimates more precise.

Question 20

What happens to the standard error of M as sample size increases?
A)It increases.
B)It decreases.
C)It stays constant.
D)The standard error does not change in a predictable manner when sample size increases.

Accepted Answer

As sample size increases, the standard error of the mean decreases. This is because as the sample size increases, the sample means become more representative of the population mean, and so the variability of the sample means around the population mean decreases, resulting in a smaller standard error.

Quiz 7: Probability and Samples: The Distribution of Sample Means.

A sample is selected from a normal population with μ = 50 and σ = 12.Which of the following samples would be considered extreme and unrepresentative for this population?

Two samples probably will have different means even if they are both the same size and they are both selected from the same population.

If samples are selected from a normal population,the distribution of sample means will also be normal.

A sample is selected from a population with a mean of µ = 40.If the sample mean is M = 45,which of the following combinations would make the sample mean a good,representative value for the population?

Samples of n = 9 scores are selected from a population.If the distribution of sample means has an expected value of 40,then the population has a mean of μ = 40.

If the sample size is equal to the population variance (n = σ²),then the standard error is equal to 1.

A sample is selected from a population with a mean of µ = 40.If the sample mean is M = 45,which of the following combinations would make the sample mean an extreme,unrepresentative value for the population?

A sample of n = 4 scores is selected from a normal population with a mean of µ = 50 and a standard deviation of σ = 20.What is the probability of obtaining a sample mean greater than M = 48?

Which combination of factors will produce the largest value for the standard error?

Samples of n = 4 scores are selected from a population.If the distribution of sample means has a standard error σ = 5 points,then the population from which the samples were obtained has a standard deviation of σ = 20.

A random sample of n = 9 scores is selected from a normal distribution with μ = 80 and σ = 12.What is the probability that the sample mean will be between 76 and 84?

A random sample of n = 16 scores is selected from a normal distribution with μ = 500 and σ = 200.For this sample,which of the following is true?

The mean for the distribution of sample means is always equal to the mean for the population from which the samples are obtained.

In order for the distribution of sample means to be normal,it must be based on samples of at least n = 30 scores.

According to the central limit theorem,the standard error for a sample mean becomes smaller as the sample size increases.

A sample of n = 25 scores is selected from a population with a mean of µ = 80 and a standard deviation of σ = 20.The standard error for the sample mean is 20.

Samples of n = 16 are selected from a population.If the distribution of sample means has a standard error of 2 points,then the population has a standard deviation of σ = 8.

A sample of n = 16 scores is selected from a population with μ = 100 and σ = 32.If the sample mean is M = 104,what is the z-score for this sample mean?

What happens to the expected value of M as sample size increases?

What happens to the standard error of M as sample size increases?