Question 1

What is the sample variance and the estimated standard error for a sample of n = 9 scores with SS = 72? A)s² = 9 and s_M = 3 B)s² = 9 and s_M = 1 C)s² = 3 and s_M = 3 D)s² = 3 and s_M = 1

Accepted Answer

The formula for sample variance is SS/n-1, so the sample variance for this problem would be 72/(9-1) = 9. 
The formula for estimated standard error is the square root of the sample variance divided by the sample size, so the estimated standard error would be sqrt(9/9) = 1. 
Option B has the correct values for both sample variance and estimated standard error.

Question 2

A sample with a mean of M = 40 and a variance of s² = 20 has an estimated standard error of 2 points.How many scores are in the sample? A)4 B)5 C)10 D)25

Accepted Answer

The estimated standard error (SE) of the mean is calculated using the formula SE = s / sqrt(n), where s is the standard deviation and n is the sample size. Given that the variance is 20, the standard deviation (s) is the square root of 20, which is approximately 4.47. With an SE of 2, we solve for n in the equation 2 = 4.47 / sqrt(n), which gives n = 5.

Question 3

If other factors are held constant,what is the effect of increasing the sample variance? A)It will increase the estimated standard error and increase the likelihood of rejecting H₀. B)It will increase the estimated standard error and decrease the likelihood of rejecting H₀. C)It will decrease the estimated standard error and increase the likelihood of rejecting H₀. D)It will decrease the estimated standard error and decrease the likelihood of rejecting H₀.

Accepted Answer

Increasing the sample variance will increase the estimated standard error, which means the sample mean is less likely to be significantly different from the null hypothesis value. Therefore, the likelihood of rejecting H₀ decreases.

Question 4

A researcher conducts a hypothesis test using a sample of n = 40 from an unknown population.What is the df value for the t statistic?​
A)​39
B)​40
C)​41
D)​42

Accepted Answer

The degrees of freedom (df) for a t statistic is calculated as the sample size (n) minus 1. Therefore, for a sample size of n = 40, the df = 40 - 1 = 39.

Question 5

Which of the following samples will have the smallest value for the estimated standard error? A)n = 25 with s² = 100 B)n = 25 with s² = 400 C)n = 100 with s² = 100 D)n = 100 with s² = 400

Accepted Answer

The estimated standard error is inversely proportional to the sample size and directly proportional to the variance. Therefore, the sample with the largest sample size and smallest variance (option C) will have the smallest estimated standard error.

Question 6

If other factors are held constant,which set of sample characteristics is most likely to reject a null hypothesis stating that m = 80?​
A)​M = 85 and small sample variance
B)​M = 85 and large sample variance
C)​M = 90 and small sample variance
D)​M = 90 and large sample variance

Accepted Answer

A larger sample mean (M = 90) than the null hypothesis value of 80 is more likely to reject the null hypothesis. However, a small sample variance may make the results less reliable. Therefore, choice C (M = 90 and small sample variance) is the most likely set of characteristics to reject the null hypothesis.

Question 7

A sample of n = 4 scores has SS = 60.What is the variance for this sample?​
A)30​
B)​20
C)​16
D)​15

Accepted Answer

The variance ($s^2$) for a sample is calculated by dividing the sum of squares (SS) by the degrees of freedom (n-1). For n = 4, degrees of freedom = 4 - 1 = 3. Thus, variance = SS / (n - 1) = 60 / 3 = 20.

Question 8

A sample of n = 4 scores has SS = 48.What is the estimated standard error for the sample mean?​
A)1​
B)​2
C)​4
D)​16

Accepted Answer

The estimated standard error for the sample mean ($SE_{\bar{x}}$) is calculated using the formula $SE_{\bar{x}} = \sqrt{\frac{SS}{n(n-1)}}$, where SS is the sum of squares and n is the sample size. Substituting the given values, $SE_{\bar{x}} = \sqrt{\frac{48}{4(4-1)}} = \sqrt{\frac{48}{12}} = \sqrt{4} = 2$.

Question 9

On average,what value is expected for the t statistic when the null hypothesis is true?​
A)0​
B)​1
C)​1.96
D)​t > 1.96

Accepted Answer

When the null hypothesis is true, we expect the sample mean to be equal to the hypothesized population mean, and the standard error to be equal to the population standard deviation divided by the square root of the sample size. In this case, the t statistic will be equal to (sample mean - hypothesized population mean)/standard error, which simplifies to (sample mean - hypothesized population mean)/0. Since division by zero is undefined, the t statistic will be 0 when the null hypothesis is true.

Question 10

When n is small (less than 30),how does the shape of the t distribution compare to the normal distribution?​
A)​It is almost perfectly normal.
B)​It is flatter and more spread out than the normal distribution.
C)​It is taller and narrower than the normal distribution.
D)​There is no consistent relationship between the t distribution and the normal distribution.

Accepted Answer

When n is small, the t distribution has heavier tails than the normal distribution, meaning that extreme values are more likely to occur. This makes the t distribution flatter and more spread out than the normal distribution.

Question 11

A sample of n = 25 scores has a mean of M = 40 and a variance of s² = 100.What is the estimated standard error for the sample mean?? A)4? B)?2 C)? $\sqrt { 2 }$ D)?1

Accepted Answer

The answer of A sample of n = 25 scores...

Question 12

A researcher conducts a hypothesis test using a sample from an unknown population.If the t statistic has df = 35,how many individuals were in the sample?​
A)​n = 34
B)​n = 35
C)​n = 36
D)​n=32

Accepted Answer

The degrees of freedom (df) for a t-test can be calculated using the formula df = n - 1, where n is the sample size. In this case, df = 35, so we can solve for n:

df = n - 1
35 = n - 1
36 = n

Therefore, the sample size was n = 36.

Question 13

Why are t statistics more variable than z-scores?​
A)​The extra variability is caused by variations in the sample mean.
B)​The extra variability is caused by variations in the sample variance.
C)​The extra variability is caused by variations in the df value.
D)​The extra variability is caused by the large sample size.

Accepted Answer

The t statistic accounts for the variability in the sample variance, which is not known a priori and must be estimated from the sample data. This estimation process introduces additional variability into the t statistic compared to the z-score, which assumes the population variance is known.

Question 14

If other factors are held constant,what is the effect of increasing the sample size? A)It will increase the estimated standard error and increase the likelihood of rejecting H₀. B)It will increase the estimated standard error and decrease the likelihood of rejecting H₀. C)It will decrease the estimated standard error and increase the likelihood of rejecting H₀. D)It will decrease the estimated standard error and decrease the likelihood of rejecting H₀.

Accepted Answer

Increasing the sample size decreases the estimated standard error, making the test statistic larger for a given effect size, thereby increasing the likelihood of rejecting the null hypothesis if it is false.

Question 15

If two samples are selected from the same population,under what circumstances are the two samples guaranteed to have exactly the same t statistic?​
A)​If the samples are the same size and have the same variance
B)​If the samples are the same size and have the same mean
C)​If the samples have the same mean and the same variance
D)​If the samples are the same size and have the same mean and have the same variance

Accepted Answer

The t statistic is dependent on sample size, sample mean, and sample variance. Therefore, if two samples are selected from the same population and have the same size, mean, and variance, they will have exactly the same t statistic.

Question 16

If a researcher is using a t statistic to test a null hypothesis about a population,what information is needed from the population to calculate the t statistic?​
A)​You must know the population mean
B)​You must know the population variance or standard deviation
C)​You must know the population mean and the variance or standard deviation
D)​The t statistic does not require any information about the population

Accepted Answer

The t statistic is used when the population standard deviation is unknown, relying instead on the sample standard deviation and mean to estimate the standard error.

Question 17

Which of the following is a fundamental difference between the t statistic and a z-score?​
A)​The t statistic uses the sample mean in place of the population mean.
B)​The t statistic uses the sample variance in place of the population variance.
C)​The t statistic computes the standard error by dividing the standard deviation by n - 1 instead of dividing by n.
D)​The t statistic is used for large samples only.

Accepted Answer

The t statistic uses the sample variance to estimate the population variance because the population variance is unknown. This is a key difference from the z-score, which uses the known population variance.

Question 18

A sample of n = 25 scores has a mean of M = 65 and an estimated standard error of 2 points.What is the sample variance? A)s² = 100 B)s² = 96 C)s² = 50 D)s² = 48

Accepted Answer

The answer of A sample of n = 25 scores...

Question 19

Which set of characteristics will produce the smallest value for the estimated standard error?​
A)​A large sample size and a small sample variance
B)​A large sample size and a large sample variance
C)​A small sample size and a small sample variance
D)​A small sample size and a large sample variance

Accepted Answer

The standard error is inversely proportional to the sample size and directly proportional to the sample variance. Therefore, a large sample size and small sample variance will produce the smallest value for the estimated standard error.

Question 20

If other factors are held constant,which set of sample characteristics is most likely to produce a significant t statistic? A)n = 25 with s² = 100 B)n = 25 with s² = 400 C)n = 100 with s² = 100 D)n = 100 with s² = 400

Accepted Answer

A significant t statistic is more likely when the sample size is larger (n = 100) and the variance (s^2) is smaller (100), as this combination reduces the standard error of the mean, making it easier to detect a significant difference if one exists.

Quiz 9: Introduction to the T Statistic

What is the sample variance and the estimated standard error for a sample of n = 9 scores with SS = 72?

A sample with a mean of M = 40 and a variance of s² = 20 has an estimated standard error of 2 points.How many scores are in the sample?

If other factors are held constant,what is the effect of increasing the sample variance?

A researcher conducts a hypothesis test using a sample of n = 40 from an unknown population.What is the df value for the t statistic?

Which of the following samples will have the smallest value for the estimated standard error?

If other factors are held constant,which set of sample characteristics is most likely to reject a null hypothesis stating that m = 80?

A sample of n = 4 scores has SS = 60.What is the variance for this sample?

A sample of n = 4 scores has SS = 48.What is the estimated standard error for the sample mean?

On average,what value is expected for the t statistic when the null hypothesis is true?

When n is small (less than 30) ,how does the shape of the t distribution compare to the normal distribution?

A sample of n = 25 scores has a mean of M = 40 and a variance of s² = 100.What is the estimated standard error for the sample mean??

A researcher conducts a hypothesis test using a sample from an unknown population.If the t statistic has df = 35,how many individuals were in the sample?

Why are t statistics more variable than z-scores?

If other factors are held constant,what is the effect of increasing the sample size?

If two samples are selected from the same population,under what circumstances are the two samples guaranteed to have exactly the same t statistic?

If a researcher is using a t statistic to test a null hypothesis about a population,what information is needed from the population to calculate the t statistic?

Which of the following is a fundamental difference between the t statistic and a z-score?

A sample of n = 25 scores has a mean of M = 65 and an estimated standard error of 2 points.What is the sample variance?

Which set of characteristics will produce the smallest value for the estimated standard error?

If other factors are held constant,which set of sample characteristics is most likely to produce a significant t statistic?

Quiz 9: Introduction to the T Statistic

What is the sample variance and the estimated standard error for a sample of n = 9 scores with SS = 72?​

A sample with a mean of M = 40 and a variance of s2 = 20 has an estimated standard error of 2 points.How many scores are in the sample?​

If other factors are held constant,what is the effect of increasing the sample variance?​

A researcher conducts a hypothesis test using a sample of n = 40 from an unknown population.What is the df value for the t statistic?​

Which of the following samples will have the smallest value for the estimated standard error?​

If other factors are held constant,which set of sample characteristics is most likely to reject a null hypothesis stating that m = 80?​

A sample of n = 4 scores has SS = 60.What is the variance for this sample?​

A sample of n = 4 scores has SS = 48.What is the estimated standard error for the sample mean?​

On average,what value is expected for the t statistic when the null hypothesis is true?​

When n is small (less than 30) ,how does the shape of the t distribution compare to the normal distribution?​

A sample of n = 25 scores has a mean of M = 40 and a variance of s2 = 100.What is the estimated standard error for the sample mean??

A researcher conducts a hypothesis test using a sample from an unknown population.If the t statistic has df = 35,how many individuals were in the sample?​

Why are t statistics more variable than z-scores?​

If other factors are held constant,what is the effect of increasing the sample size?​

If two samples are selected from the same population,under what circumstances are the two samples guaranteed to have exactly the same t statistic?​

If a researcher is using a t statistic to test a null hypothesis about a population,what information is needed from the population to calculate the t statistic?​

Which of the following is a fundamental difference between the t statistic and a z-score?​

A sample of n = 25 scores has a mean of M = 65 and an estimated standard error of 2 points.What is the sample variance?​

Which set of characteristics will produce the smallest value for the estimated standard error?​

If other factors are held constant,which set of sample characteristics is most likely to produce a significant t statistic?​