Deck 9: Introduction to the T Statistic

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

Answer 1

​s² = 9 and s_M = 1

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

Answer 2

​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.

Answer 3

​It will increase the estimated standard error and decrease the likelihood of rejecting H₀.

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

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

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

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

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

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

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.

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

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

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.

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.

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

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

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.

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

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

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

Question 21

With α = .05 and df = 8,the critical values for a two-tailed t test are t = ±2.306.Assuming all other factors are held constant,if the df value were increased to df = 20,what would happen to the critical values for t?​

A)​They would increase (move farther from zero).
B)​They would decrease (move closer to zero).
C)​They would stay the same.
D)​There is not enough information to answer.

Question 22

If other factors are held constant,how does sample size influence the likelihood of rejecting the null hypothesis and measures of effect size such as r² and Cohen's d?​

A)​A larger sample increases both the likelihood and measures of effect size.
B)​A larger sample increases the likelihood but has little influence on measures of effect size.
C)​A larger sample decreases the likelihood but has little influence on measures of effect size.
D)​A larger sample decreases both the likelihood and measures of effect size.

Question 23

Under what circumstances can a very small treatment effect be statistically significant?​

A)​If the sample size is big and the sample variance is small
B)​If the sample size and the sample variance are both big
C)​If the sample size is small and the sample variance is big
D)​If the sample size and the sample variance are both small

Question 24

What value is estimated with a confidence interval using the t statistic?​

A)​The value for an unknown sample mean
B)​The value for an unknown population mean
C)​The difference between two population means
D)​The difference between two sample means

Question 25

How does sample variance influence the likelihood of rejecting the null hypothesis and measures of effect size such as r² and Cohen's d?​

A)​Larger variance increases both the likelihood and measures of effect size.
B)​Larger variance increases the likelihood but decreases measures of effect size.
C)​Larger variance decreases the likelihood but increases measures of effect size.
D)​Larger variance decreases both the likelihood and measures of effect size.

Question 26

A sample of n = 4 scores is selected from a population with an unknown mean.The sample has a mean of M = 40 and a variance of s² = 16.Which equation correctly describes the 90% confidence interval for μ?​

A)​μ = 40 ± 2.353(4)
B)​μ = 40 ± 1.638(4)
C)​μ = 40 ± 2.353(2)
D)​μ = 40 ± 1.638(2)

Question 27

A sample has a mean of M = 39.5 and a standard deviation of s = 4.3,and produces a t statistic of t = 2.14.For a two-tailed hypothesis test with α = .05,what is the correct statistical decision for this sample?​

A)​The researcher can reject the null hypothesis with α = .05 but not with α = .01.
B)​The researcher can reject the null hypothesis with either α = .05 or α = .01.
C)​The researcher must fail to reject the null hypothesis with either α = .05 or α = .01.
D)​It is impossible to make a decision about H₀ without more information.

Question 28

With α = .05,what is the critical t value for a one-tailed test with n = 15?​

A)​t = 1.761
B)​t = 1.753
C)​t = 2.145
D)​t = 2.131

Question 29

​A sample is selected from a population with μ = 46 and a treatment is administered to the sample.After treatment,the sample mean is M = 48 with a sample variance of s² = 16.Based on this information,the size of the treatment effect,as measured by Cohen's d,is ____.

A)d = 0.125​
B)​d = 0.25
C)​d = 0.50
D)​d=0.001

Question 30

A sample of n = 25 scores produces a t statistic of t = 2.062.If the researcher is using a two-tailed test,then which of the following is the correct statistical decision?​

A)​The researcher can reject the null hypothesis with α = .05 but not with α = .01.
B)​The researcher can reject the null hypothesis with either α = .05 or α = .01.
C)​The researcher must fail to reject the null hypothesis with either α = .05 or α = .01.
D)​It is impossible to make a decision about H₀ without more information.

Question 31

A hypothesis test produces a t statistic of t = 2.20.If the researcher is using a two-tailed test with α = .05,how large does the sample have to be in order to reject the null hypothesis?​

A)​At least n = 11
B)​At least n = 12
C)​At least n = 13
D)​At least n = 14

Question 32

Two samples from the same population both have n = 10 scores with M = 45.If the t statistic is computed for each sample,then what is the relationship between the two t values?​

A)​The two t statistics will be identical.
B)​The sample with the larger variance will produce the larger t statistic.
C)​The sample with the smaller variance will produce the larger t statistic.
D)​There is no way to predict the relationship between the two t statistics

Question 33

A sample of n = 16 scores produces a t statistic of t = 2.00.If the sample is used to measure effect size with r²,what value will be obtained for r²?​

A)​r² = 2/20
B)​r² = 4/20
C)​r² = 2/19
D)​r² = 4/19

Question 34

Which of the following describes the function of a confidence interval?​

A)​It uses a sample mean to estimate the corresponding population mean.
B)​It uses a population mean to predict a sample mean.
C)​It uses a level of confidence to estimate a sample mean.
D)​It uses the sample mean to determine a level of confidence.

Question 35

With α = .01 the two-tailed critical region for a t test using a sample of n = 16 subjects would have boundaries of ____.​

A)​t = ±2.602
B)​t = ±2.583
C)​t = ±2.947
D)​t = ±2.921

Question 36

A hypothesis test with a sample of n = 25 participants produces a t statistic of t = +2.53.Assuming a one-tailed test with the critical region in the right-hand tail,what is the correct decision?​

A)​The researcher can reject the null hypothesis with α = .05 but not with α = .01.
B)​The researcher can reject the null hypothesis with either α = .05 or α = .01.
C)​The researcher must fail to reject the null hypothesis with either α = .05 or α = .01.
D)​It is impossible to make a decision about H₀ without more information.

Question 37

The results of a hypothesis test are reported as follows: t(15)= 2.70,p < .05.Based on this report,how many individuals were in the sample?​

A)​14
B)​15
C)​16
D)​17

Question 38

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 for a sample of n = 25
B)​M = 85 for a sample of n = 100
C)​M = 90 for a sample of n = 25
D)​M = 90 for a sample of n = 100

Question 39

If a sample of n = 16 scores is being used to make an 80% confidence interval estimate of the population mean,μ,what value(s)of t should be used?​

A)​t = 0
B)​t = ±2.131
C)​t = ±1.753
D)​t = ±1.341

Question 40

When the population variance or standard deviation is not known,you must use a t statistic instead of a z-score for a hypothesis test.

Question 41

If two samples,each with n = 20 scores,are selected from the same population and both have the same mean (M = 53)and the same variance (s² = 12),then they will also have the same t statistic.

Question 42

If random samples,each with n = 20 scores,are selected from a population,and the z-score and t statistic are computed for each sample,the t statistics will be more variable than the z-scores.

Question 43

Compared to a z-score,a hypothesis test with a t statistic requires more information from the sample.

Question 44

If other factors are held constant,the bigger the sample is,the greater the likelihood of rejecting the null hypothesis.

Question 45

A sample of n = 16 scores with a sample variance of s² = 64 would have an estimated standard error of 4 points.

Question 46

As the sample size is increased,the distribution of t statistics becomes flatter and more spread out.

Question 47

If other factors are held constant,as the sample size increases,the estimated standard error decreases.

Question 48

If a hypothesis test using a sample of n = 16 scores produces a t statistic of t = 2.15,then the correct decision is to reject the null hypothesis for a two-tailed test with a = .05.

Question 49

For a one-tailed test with α = .05 and a sample of n = 9,the critical value for the t statistic is t = 1.860.

Question 50

The t distribution for df = 4 is flatter and more spread out than the t distribution for df = 20.

Question 51

Two samples from same population probably will have different t statistics even if they are the same size and have the same mean.

Question 52

For a two-tailed test with α = .05 and a sample of n = 16,the boundaries for the critical region are t = ±2.120.

Question 53

If two samples each have the same mean,the same number of scores,and are selected from the same population,then they will also have identical t statistics.

Question 54

A sample of n = 4 scores with SS = 48 has a variance of 16 and an estimated standard error of 2.

Question 55

In general,the larger the value of the sample variance,the greater the likelihood of rejecting the null hypothesis.

Question 56

For a two-tailed hypothesis test with α = .05 and a sample of n = 25 scores,the boundaries for the critical region are t = ±2.060.

Question 57

If other factors are held constant,as the sample variance increases,the estimated standard error also increases.

Question 58

For a hypothesis test using a t statistic,the boundaries for the critical region will change if the sample size is changed.

Question 59

As sample size increases,the critical region boundaries for a two-tailed test with a = .05 will move closer to zero.

Question 60

In a hypothesis test,a large value for the sample variance increases the likelihood that you will find a significant treatment effect.

Question 61

Two samples are selected from a population and a treatment is administered to the samples.If both samples have the same mean and the same variance,you are more likely to find a significant treatment effect with a sample of n = 100 than with a sample of n = 4.

Question 62

For a one tailed test evaluating a treatment that is supposed to decrease scores,a researcher obtains t(8)= 1.90.For α = .05,the correct decision is to reject the null hypothesis.

Question 63

A sample of n = 16 scores has a mean of M = 58 with SS = 960.Use the sample to construct the 90% confidence interval for µ.

Question 64

If the 90% confidence interval for µ is from 40 to 50,then the sample mean is M = 45.

Question 65

In order to estimate a population mean with a confidence interval,you first must estimate a range of values for t.

Question 66

A research report states "t(8)= 2.00,p > .05." For this test,r² = 2/10.

Question 67

If the null hypothesis states that m = 70 and a researcher obtains a sample with M = 73 and s² = 9,then Cohen's d = 0.33.

Question 68

A researcher is testing the effect of a new cold and flu medication on mental alertness.A sample of n = 9 college students is obtained and each student is given the normal dose of the medicine.Thirty minutes later,each student's performance is measured on a video game that requires careful attention and quick decision making.The scores for the nine students are as follows: 6,8,10,6,7,13,5,5,3.
a. Assuming that scores for students in the regular population average μ = 10, are the data sufficient to conclude that the medication has a significant effect on mental performance? Test at the .05 level of significance.
b. Compute r2, the percentage of variance explained by the treatment effect.
c. Write a sentence demonstrating how the outcome of the hypothesis test and the measure of effect size would be presented in a research report.

Question 69

The sample mean will always be exactly in the center of a confidence interval that is estimating the value of the population mean.

Question 70

Although hypothesis tests are affected by sample size,it has little or no influence on measures of effect size,such as r² or Cohen's
d.

Question 71

A sample of n = 16 individuals is selected from a population with µ = 30.After a treatment is administered to the individuals,the sample mean is found to be M = 33.
a. If the sample variance is s2 = 16, then calculate the estimated standard error and determine whether the sample is sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with α = .05.
b. If the sample variance is s2 = 64, then calculate the estimated standard error and determine whether the sample is sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with α = .05.
c. Describe how increasing variance affects the standard error and the likelihood of rejecting the null hypothesis.

Question 72

A sample is selected from a population with µ = 80.After a treatment is administered to the individuals,the sample mean is found to be M = 75 and the variance is s2 = 100.
a. If the sample has n = 4 scores, then calculate the estimated standard error and determine whether the sample is sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with α = .05.
b. If the sample has n = 25 scores, then calculate the estimated standard error and determine whether the sample is sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with α = .05.
c. Describe how increasing the size of the sample affects the standard error and the likelihood of rejecting the null hypothesis.

Question 73

Although you can compute a z-score for a single score (a sample of n = 1),it is impossible to compute a t statistic for a sample that has only one score.Explain why.

Question 74

A research report states "t(15)= 2.31,p < .05." For this study,the sample had n = 16 scores.