Question 1

For a normal population with µ = 40 and σ = 10 which of the following samples is least likely to be obtained?
A)M ≤38 for a sample of n = 4
B)M ≤36 for a sample of n = 4
C)M ≤38 for a sample of n = 100
D)M ≤36 for a sample of n = 100

Accepted Answer

As the sample size increases, the variability of the sampling distribution decreases, which means that extreme values are less likely to be obtained. Therefore, the sample with n=100 is more likely to have a mean closer to the population mean of 40. In addition, a mean of 36 is more than 2 standard deviations below the population mean, so it is a very low probability event for a normal distribution with a standard deviation of 10. Therefore, option D is the least likely sample to be obtained.

Question 2

A sample of n = 16 scores is obtained from a population with μ = 50 and σ = 16.If the sample mean is M = 54,then what is the z-score for the sample mean?
A)z = 4.00
B)z = 1.00
C)z = 0.50
D)z = 0.25

Accepted Answer

To calculate the z-score for the sample mean, we use the formula:

z = (M - μ) / (σ / sqrt(n))

Plugging in the values given, we get:

z = (54 - 50) / (16 / sqrt(16)) = 4 / 4 = 1

Therefore, the z-score for the sample mean is 1, which corresponds to choice B.

Question 3

A random sample of n = 16 scores is obtained from a normal population with µ = 40 and σ = 8.What is the probability that the sample mean will be within 2 points of the population mean?
A)0.3830
B)0.6826
C)0.8664
D)0.9544

Accepted Answer

The probability that the sample mean will be within 2 points of the population mean can be found using the 68-95-99.7 rule (Empirical Rule) for normal distributions. Since we are looking for the probability within 2 points (which is within 1 standard deviation of the mean for the sampling distribution), the probability is approximately 68.26% or 0.6826. The standard error of the mean (σ/√n) is used to determine the standard deviation of the sampling distribution, which in this case is 8/√16 = 2. This aligns with being within 1 standard deviation of the mean in the context of the sampling distribution.

Question 4

For a normal population with a mean of µ = 80 and a standard deviation of σ = 10,what is the probability of obtaining a sample mean greater than M = 75 for a sample of n = 25 scores?
A)0.0062
B)0.9938
C)0.3085
D)0.6915

Accepted Answer

To find the probability of obtaining a sample mean greater than 75, we use the formula for the standard error of the mean (σM = σ / √n) which in this case is 10 / √25 = 2. Then, we calculate the z-score for M = 75, which is (75 - 80) / 2 = -2.5. Looking up this z-score in a standard normal distribution table or using a calculator, we find the probability of a z-score less than -2.5. Since we want greater than 75, we look for the complement of this, which gives us 0.9938.

Question 5

A sample of n = 4 scores has a standard error of 10 points.For the same population,what is the standard error for a sample of n = 16 scores?
A)1
B)2.5
C)5
D)10

Accepted Answer

The formula for calculating standard error is:

SE = s / sqrt(n)

where s is the sample standard deviation and n is the sample size. We are given that the standard error for n=4 is 10, therefore:

10 = s / sqrt(4)

=> s = 20

Now, we need to find the standard error for n=16. Using the same formula:

SE = s / sqrt(n) = 20 / sqrt(16) = 5

Therefore, the standard error for a sample of n=16 scores is 5.

Question 6

If a sample of n = 4 scores is obtained from a population with μ = 70 and σ = 12,what is the z-score corresponding to a sample mean of M = 73?
A)z = 0.25
B)z = 0.50
C)z = 1.00
D)z = 2.00

Accepted Answer

The z-score is calculated using the formula: z = (M - μ) / (σ/√n). Substituting the given values: z = (73 - 70) / (12/√4) = 3 / 6 = 0.50.

Question 7

A sample from a population with μ = 40 and σ = 8 has a mean of M = 36.If the sample mean corresponds to a z = -1.00,then how many scores are in the sample?
A)n = 64
B)n = 16
C)n = 8
D)n = 4

Accepted Answer

The z-score formula for a sample mean is z = (M - μ) / (σ/√n). Plugging in the values: -1 = (36 - 40) / (8/√n). Solving for n gives n = 4.

Question 8

A sample is obtained from a population with μ = 100 and σ = 20.Which of the following samples would produce the most extreme z-score?
A)A sample of n = 25 scores with M = 102
B)A sample of n = 100 scores with M = 102
C)A sample of n = 25 scores with M = 104
D)A sample of n = 100 scores with M = 104

Accepted Answer

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

Question 9

A sample of n = 9 scores is obtained from a population with μ = 70 and σ = 18.If the sample mean is M = 76,what is the z-score for the sample mean?
A)z = 0.33
B)z = 0.50
C)z = 1.00
D)z = 3.00

Accepted Answer

To calculate the z-score for the sample mean, we will use the formula:

z = (M - μ) / (σ / √n)

Here, M = 76, μ = 70, σ = 18, and n = 9. Substituting these values, we get:

z = (76 - 70) / (18 / √9) = 6 / 6 = 1.00

Therefore, the z-score for the sample mean is 1.00.

Question 10

A sample of n = 4 scores is obtained from a population with μ = 70 and σ = 8.If the sample mean corresponds to a z-score of 2.00,what is the value of the sample mean?
A)M = 86
B)M = 78
C)M = 74
D)M = 72

Accepted Answer

The z-score formula when applied to a sample mean is z = (M - μ) / (σ/√n), where M is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Given z = 2.00, μ = 70, σ = 8, and n = 4, we solve for M: 2 = (M - 70) / (8/√4) => 2 = (M - 70) / 4 => M = 78.

Question 11

For a normal population with µ = 40 and σ = 10,which of the following samples has the highest probability of being obtained?
A)M ≤38 for a sample of n = 4
B)M≤ 36 for a sample of n = 4
C)M ≤38 for a sample of n = 100
D)M 36 for a sample of n = 100

Accepted Answer

The standard error is larger for smaller samples, making it more likely to obtain sample means further from the population mean. Thus, M ≤ 38 for n = 4 is more probable than M ≤ 36 for n = 4.

Question 12

A sample is selected from a population with μ = 50 and σ = 12.If the sample mean of M = 56 produces a z-score of z = +1.00,then how many scores are in the sample?
A)2
B)4
C)9
D)16

Accepted Answer

The z-score formula is z = (M - μ) / (σ/√n). Plugging in the values: 1 = (56 - 50) / (12/√n). Solving for n gives n = 4.

Question 13

A sample of n = 36 scores is selected from a population with σ = 12.If the sample mean of M = 56 produces a z-score of z = +3.00,then what is the population mean?
A)50
B)52
C)54
D)56

Accepted Answer

The population mean (μ) can be found using the formula for the z-score: $z = \frac{M - \mu}{\sigma / \sqrt{n}}$, where M is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Plugging in the given values: $3 = \frac{56 - \mu}{12 / \sqrt{36}}$, solving for μ gives μ = 50.

Question 14

A random sample of n = 9 scores is obtained from a normal population with µ = 40 and σ = 18.What is the probability that the sample mean will be greater than M = 43?
A)0.4325
B)0.3085
C)0.1587
D)0.0228

Accepted Answer

The probability that the sample mean will be greater than 43 can be found using the z-score formula for a sample mean: $z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}$, where $\bar{x}$ 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: $z = \frac{43 - 40}{18/\sqrt{9}} = \frac{3}{6} = 0.5$. Looking up 0.5 on the standard normal distribution table or using a calculator gives a right-tail probability of approximately 0.3085.

Question 15

A random sample of n = 4 scores is obtained from a population with a mean of µ = 80 and a standard deviation of σ = 10.If the sample mean is M = 90,what is the z-score for the sample mean?
A)z = 20.00
B)z = 5.00
C)z = 2.00
D)z = 1.00

Accepted Answer

The formula for calculating the z-score of a sample mean is z = (M - µ) / (σ / sqrt(n)). Plugging in the values given, we get z = (90 - 80) / (10 / sqrt(4)) = 2. Therefore, the z-score for the sample mean is 2.

Question 16

A sample obtained from a population with σ = 12 has a standard error of 2 points.How many scores are in the sample?
A)n = 36
B)n = 24
C)n = 6
D)n = 3

Accepted Answer

The standard error (SE) is calculated using the formula SE = σ / √n, where σ is the population standard deviation and n is the sample size. Given SE = 2 and σ = 12, solving for n gives n = (12 / 2)^2 = 36.

Question 17

A sample of n = 16 scores has a standard error of 4.What is the standard deviation of the population from which the sample was obtained?
A)64
B)16
C)4
D)2

Accepted Answer

The standard error (SE) is calculated as the population standard deviation (σ) divided by the square root of the sample size (n). Given SE = 4 and n = 16, we have 4 = σ / √16. Solving for σ gives σ = 4 * 4 = 16.

Question 18

A random sample of n = 16 scores is obtained from a population with σ = 12.If the sample mean is 6 points greater than the population mean,what is the z-score for the sample mean?
A)+6.00
B)+2.00
C)+1.00
D)Cannot be determined without knowing the population mean

Accepted Answer

We can use the formula 
z = (x̄ - μ) / (σ/√n)
where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Since we know that the sample mean is 6 points greater than the population mean, we can write:
x̄ = μ + 6

Substituting this into the formula, we get:
z = (μ + 6 - μ) / (12/√16)
z = 2

Therefore, the z-score for the sample mean is +2.00.

Question 19

A random sample of n = 4 scores is obtained from a normal population with µ = 30 and σ = 8.What is the probability that the sample mean will be smaller than M = 22?
A)0.00003
B)0.0228
C)0.1587
D)0.3085

Accepted Answer

The probability that the sample mean will be smaller than M = 22 can be found using the z-score formula for a sample mean: $z = \frac{M - \mu}{\sigma/\sqrt{n}}$, 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: $z = \frac{22 - 30}{8/\sqrt{4}} = \frac{-8}{4} = -2$. Looking up a z-score of -2 in the standard normal distribution table gives a probability of approximately 0.0228.

Question 20

A sample of n = 4 scores is selected from a population with μ = 50 and σ = 12.If the sample mean is M = 56,what is the z-score for this sample mean?
A)0.50
B)1.00
C)2.00
D)4.00

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: $z = \frac{56 - 50}{12/\sqrt{4}} = \frac{6}{6} = 1.00$.

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

For a normal population with µ = 40 and σ = 10 which of the following samples is least likely to be obtained?

A sample of n = 16 scores is obtained from a population with μ = 50 and σ = 16.If the sample mean is M = 54,then what is the z-score for the sample mean?

A random sample of n = 16 scores is obtained from a normal population with µ = 40 and σ = 8.What is the probability that the sample mean will be within 2 points of the population mean?

For a normal population with a mean of µ = 80 and a standard deviation of σ = 10,what is the probability of obtaining a sample mean greater than M = 75 for a sample of n = 25 scores?

A sample of n = 4 scores has a standard error of 10 points.For the same population,what is the standard error for a sample of n = 16 scores?

If a sample of n = 4 scores is obtained from a population with μ = 70 and σ = 12,what is the z-score corresponding to a sample mean of M = 73?

A sample from a population with μ = 40 and σ = 8 has a mean of M = 36.If the sample mean corresponds to a z = -1.00,then how many scores are in the sample?

A sample is obtained from a population with μ = 100 and σ = 20.Which of the following samples would produce the most extreme z-score?

A sample of n = 9 scores is obtained from a population with μ = 70 and σ = 18.If the sample mean is M = 76,what is the z-score for the sample mean?

A sample of n = 4 scores is obtained from a population with μ = 70 and σ = 8.If the sample mean corresponds to a z-score of 2.00,what is the value of the sample mean?

For a normal population with µ = 40 and σ = 10,which of the following samples has the highest probability of being obtained?

A sample is selected from a population with μ = 50 and σ = 12.If the sample mean of M = 56 produces a z-score of z = +1.00,then how many scores are in the sample?

A sample of n = 36 scores is selected from a population with σ = 12.If the sample mean of M = 56 produces a z-score of z = +3.00,then what is the population mean?

A random sample of n = 9 scores is obtained from a normal population with µ = 40 and σ = 18.What is the probability that the sample mean will be greater than M = 43?

A random sample of n = 4 scores is obtained from a population with a mean of µ = 80 and a standard deviation of σ = 10.If the sample mean is M = 90,what is the z-score for the sample mean?

A sample obtained from a population with σ = 12 has a standard error of 2 points.How many scores are in the sample?

A sample of n = 16 scores has a standard error of 4.What is the standard deviation of the population from which the sample was obtained?

A random sample of n = 16 scores is obtained from a population with σ = 12.If the sample mean is 6 points greater than the population mean,what is the z-score for the sample mean?

A random sample of n = 4 scores is obtained from a normal population with µ = 30 and σ = 8.What is the probability that the sample mean will be smaller than M = 22?

A sample of n = 4 scores is selected from a population with μ = 50 and σ = 12.If the sample mean is M = 56,what is the z-score for this sample mean?