Question 1

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)​It cannot be determined without knowing the population mean.

Accepted Answer

To find the z-score of the sample mean, we need to calculate the standard error of the mean using the formula: 
SE = σ/√n
where σ is the population standard deviation and n is the sample size.
SE = 12/√16 = 3
The sample mean is 6 points greater than the population mean, so we can write: 
x̄ - μ = 6
where x̄ is the sample mean and μ is the population mean. Rearranging this equation, we have: 
x̄ = μ + 6
Substituting the values, we get: 
x̄ - μ = 6
(μ + 6) - μ = 6
6 = 6
This means that the sample mean is exactly 6 points greater than the population mean. Therefore, the z-score for the sample mean is: 
z = (x̄ - μ)/SE = 6/3 = 2
So, the answer is B (z-score of 2).

Question 2

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)p = 0.0062​
B)​p = 0.9938
C)​p = 0.3085
D)​p = 0.6915

Accepted Answer

To find the probability of obtaining a sample mean greater than 75, we use the Z-score formula for a sample mean: 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. Plugging in the values: Z = (75 - 80) / (10/√25) = -5 / (10/5) = -5 / 2 = -2.5. Looking up the Z-score of -2.5 in the Z-table gives the probability of obtaining a score less than 75. To find the probability of obtaining a score greater than 75, we subtract this value from 1. The Z-table value for -2.5 is approximately 0.0062, so 1 - 0.0062 = 0.9938, indicating a high probability of obtaining a sample mean greater than 75.

Question 3

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 z-score is calculated as (M - μ) / (σ/√n). For option D, the sample mean (M) is 104, which is the furthest from the population mean (μ = 100), and the sample size (n = 100) is the largest, resulting in the smallest standard error (σ/√n), thus producing the most extreme z-score.

Question 4

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,then what is the value of the sample mean?​
A)​M = 86
B)​M = 78
C)​M = 74
D)​M = 72

Accepted Answer

The formula to calculate the sample mean from a z-score is M = μ + z(σ/√n). Given μ = 70, σ = 8, n = 4, and z = 2.00, we substitute these values into the formula: M = 70 + 2(8/√4) = 70 + 2(4) = 70 + 8 = 78.

Question 5

A sample of n = 9 scores is obtained from a population with μ = 70 and σ = 18.If the sample mean is M = 76,then 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 find the z-score for the sample mean, we use the formula:

z = (M - μ) / (σ / √n)

Substituting the given values, we get:

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

Therefore, the z-score for the sample mean is z = 1.00, which corresponds to choice C.

Question 6

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 = 0.25
B)​z = 0.50
C)​z = 1.00
D)​z = 4.00

Accepted Answer

To calculate the z-score, we use the formula:
z = (M - μ) / (σ / sqrt(n))
Substitute the values:
z = (54 - 50) / (16 / sqrt(16)) = 4 / 4 = 1
Therefore, the z-score for the sample mean is 1, which is option C.

Question 7

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 σ = 12 and SE = 2, solving for n gives n = (σ / SE)^2 = (12 / 2)^2 = 36.

Question 8

For a particular population a sample of n = 4 scores has an expected mean value of 10.For the same population,a sample of n = 25 scores would have an expected mean value of ____.​
A)4​
B)​8
C)​10
D)​20

Accepted Answer

The expected mean value for a population remains the same, irrespective of the sample size. Therefore, the expected mean value for a sample of n = 25 scores from the same population would also be 10.

Question 9

For a normal population with µ = 40 and σ = 10 which of the following samples has the highest probability of being obtained?​
A)​M = 42 for a sample of n = 4
B)​M = 44 for a sample of n = 4
C)​M = 42 for a sample of n = 100
D)​M = 44 for a sample of n = 100

Accepted Answer

The answer of For a normal population with µ =...

Question 10

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 standard error decreases as the sample size increases, given that the standard deviation remains constant. The formula for standard error is SE = SD/sqrt(n), where SD is the standard deviation and n is the sample size. Therefore, if n increases four times from 4 to 16, the standard error will decrease by a factor of sqrt(4) = 2. Hence, the standard error for a sample of n = 16 scores will be 10/2 = 5 points. Therefore, the correct choice is C.

Question 11

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

We can start by finding the standard error of the mean (SEM), which is:

SEM = σ/√n = 8/√16 = 2

Next, we can use the z-score formula to find the probability of the sample mean being within 2 points of the population mean:

z = (sample mean - population mean) / SEM

z = (x̄ - µ) / SEM = (40 - 40) / 2 = 0

To find the probability, we need to look up the area under the normal distribution curve between z = -2 and z = 2. This can be done using a standard normal distribution table or calculator, and the result is:

P(-2 < z < 2) = 0.6826

Therefore, the probability that the sample mean will be within 2 points of the population mean is 0.6826, which corresponds to choice B.

Question 12

A sample of n = 9 scores has a standard error of 6.What is the standard deviation of the population from which the sample was obtained?​
A)54​
B)​18
C)​6
D)​2

Accepted Answer

The formula for standard error is:

standard error = standard deviation / square root of sample size

In this case, we know that the standard error is 6 and the sample size is 9. Solving for the standard deviation, we get:

6 = standard deviation / √9

6 = standard deviation/3

Standard deviation = 6 * 3 = 18

Therefore, the correct answer is B) 18.

Question 13

If a sample of n = 4 scores is obtained from a population with μ = 70 and σ = 12,then what is the z-score corresponding to a sample mean of M = 76?​
A)​z = 0.25
B)​z = 0.50
C)​z = 1.00
D)​z = 2.00

Accepted Answer

The answer of If a sample of n = 4...

Question 14

A sample from a population with μ = 40 and σ = 10 has a mean of M = 44.If the sample mean corresponds to a z = 2.00,then how many scores are in the sample?​
A)​n = 100
B)​n = 25
C)​n = 5
D)​n = 4

Accepted Answer

The formula to calculate the z-score is $z = \frac{M - \mu}{\sigma / \sqrt{n}}$, where $M$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size. Rearranging the formula to solve for $n$, we get $n = \left(\frac{\sigma \cdot z}{M - \mu}\right)^2$. Plugging in the given values ($\sigma = 10$, $z = 2.00$, $M = 44$, $\mu = 40$), we find $n = \left(\frac{10 \cdot 2}{44 - 40}\right)^2 = 25$.

Question 15

For a normal population with µ = 40 and σ = 10 which of the following samples is least likely to be obtained?​
A)​M = 42 for a sample of n = 4
B)​M = 44 for a sample of n = 4
C)​M = 42 for a sample of n = 100
D)​M = 44 for a sample of n = 100

Accepted Answer

The answer of For a normal population with µ =...

Question 16

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 for the sample mean is z = (M - µ) / (σ / sqrt(n)). Substituting the values given in the question, we get z = (90 - 80) / (10 / sqrt(4)) = 2. Therefore, the correct answer is option C, z = 2.00.

Question 17

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 18

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

Question 19

A random sample of n = 4 scores is obtained from a normal population with µ = 20 and σ = 4.What is the probability that the sample mean will be greater than M = 22?​
A)​0.50
B)​1.00
C)​0.1587
D)​0.3085

Accepted Answer

The standard error of the mean (SEM) is σ/√n = 4/√4 = 2. The z-score for M = 22 is (22 - 20) / 2 = 1. Using the standard normal distribution table, the probability of a z-score greater than 1 is 0.1587.

Question 20

A random sample of n = 9 scores is obtained from a normal population with µ = 40 and σ = 6.What is the probability that the sample mean will be greater than M = 43?​
A)0.3085​
B)​0.6915
C)​0.9332
D)​0.0668

Accepted Answer

Use the formula for the standard error of the mean: 
SE = σ/√n = 6/√9 = 2 
Then, convert the sample mean to a z-score: 
z = (43 - 40)/2 = 1.5 
Finally, use a z-table or calculator to find the probability of a z-score greater than 1.5, which is 0.0668.

Quiz 7: Probability and Samples: the Distribution of Sample Means

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?​

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 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 = 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,then what is the value of the sample mean?​

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

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

For a particular population a sample of n = 4 scores has an expected mean value of 10.For the same population,a sample of n = 25 scores would have an expected mean value of ____.​

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

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 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 sample of n = 9 scores has a standard error of 6.What is the standard deviation of the population from which the sample was obtained?​

If a sample of n = 4 scores is obtained from a population with μ = 70 and σ = 12,then what is the z-score corresponding to a sample mean of M = 76?​

A sample from a population with μ = 40 and σ = 10 has a mean of M = 44.If the sample mean corresponds to a z = 2.00,then how many scores are in the sample?​

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

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 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 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 random sample of n = 4 scores is obtained from a normal population with µ = 20 and σ = 4.What is the probability that the sample mean will be greater than M = 22?​

A random sample of n = 9 scores is obtained from a normal population with µ = 40 and σ = 6.What is the probability that the sample mean will be greater than M = 43?​

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?

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 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 = 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,then what is the value of the sample mean?

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

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

For a particular population a sample of n = 4 scores has an expected mean value of 10.For the same population,a sample of n = 25 scores would have an expected mean value of ____.

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

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 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 sample of n = 9 scores has a standard error of 6.What is the standard deviation of the population from which the sample was obtained?

If a sample of n = 4 scores is obtained from a population with μ = 70 and σ = 12,then what is the z-score corresponding to a sample mean of M = 76?

A sample from a population with μ = 40 and σ = 10 has a mean of M = 44.If the sample mean corresponds to a z = 2.00,then how many scores are in the sample?

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

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 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 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 random sample of n = 4 scores is obtained from a normal population with µ = 20 and σ = 4.What is the probability that the sample mean will be greater than M = 22?

A random sample of n = 9 scores is obtained from a normal population with µ = 40 and σ = 6.What is the probability that the sample mean will be greater than M = 43?