ECO 339 Exam 2 (11 & 12)
To test for the difference between the means of two populations when the standard deviations of the two populations are unknown and it can be assumed the two populations have equal variances, we must use a t distribution with (let n1 be the size of sample 1 and n2 the size of sample 2)
(n1 + n2 - 2) degrees of freedom
Which of the following statements are true?
***All of these statements are true** None of these statements is true. The z-test can be used as a close approximation to the unequal-variances t-test when the population standard deviations are not assumed to be equal, but samples are large (each n ≥ 30). The unequal-variances t-test is used whenever the population standard deviations are unknown and cannot be assumed to be equal. The pooled-variances t-test is used whenever the population standard deviations are assumed to be equal regardless of the sample size.
We are interested in testing the following hypotheses. H0: m1- m2 >= 0 Ha: m1- m2 < 0 The test statistic Z is computed to be 2.83. The p-value for this test is
.0023
When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population means, but your statistical software provides only a one-tail area of 0.03 as part of its output. The p-value for this two-tail test will be:
.06
In testing whether the means of two normal populations are equal, summary statistics computed for two independent samples are as follows: Brand X Brand Y n1=20 n2=20 x1¯¯¯¯=7.30 x2¯¯¯¯=6.80 s1=1.10 s1=1.15 Assume that the population variances are equal. Then, the standard error of the sampling distribution of the sample mean difference x¯1−x¯2 is equal to:
.3558
When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population means, but your statistical software provides only a one-tail area of 0.03 as part of its output. The p-value for this two-tail test will be
0.06
In testing whether the means of two normal populations are equal, summary statistics computed for two independent samples are as follows: Brand X Brand Y n1=20 n2=20 x1=7.30 x2=6.80 s1=1.10 s1=1.15 Assume that the population variances are equal. Then, the standard error of the sampling distribution of the sample mean difference x¯1−x¯2 is equal to
0.3558
Salary information regarding male and female employees of a large company is shown below. Male Female Sample Size M: 64 F: 36 Sample Mean Salary (in $1,000) M: 44 F: 41 Population Variance (σ2) M: 128 F: 72 If you are interested in testing whether or not the average salary of males is significantly greater than that of females, the test statistic is
1.5
An independent-measures study comparing two treatments produces a t-statistic with df = 18. If the two samples taken from populations with equal variances, are the same size, how many participants were in each of the samples?
10
In a one-way ANOVA involving five treatments, 13 observations were recorded for each of the five treatments (a total of 65 observations). The following information is provided. SSTR = 200 (Sum Square Between Treatments) SST = 800 (Total Sum Square) The mean square within treatments (MSE) is
10
SSTR = 6,750 H0: m1 = m2 = m3 = m4 SSE = 8,000 Ha: at least one mean is different n1=n2=n3=n4= 20 The mean square within treatments (MSE) equals
105.26
SSTR = 6,750 H0: m1 = m2 = m3 = m4 SSE = 8,000 Ha: at least one mean is different n1=n2=n3=n4= 20 The mean square within treatments (MSE) equals
105.26
A researcher reports an F-ratio with df = 1, 24 for an independent-measures experiment. If all the treatments had the same number of participants, then how many individuals were in each treatment?
13
One-way ANOVA is applied to independent samples taken from three normally distributed populations with equal variances. The following summary statistics were calculated: n1 = 8 x¯1= 15 s1 = 2 n2 = 10 x¯2= 18 s2 = 3 n3 = 8 x¯3= 20 s3 = 2 The within-treatments variation equals:
137
In an analysis of variance problem involving 3 treatments and 10 observations per treatment, SSE = 399.6. The MSE for this situation is
14.8
An analysis of variance produces SSbetween = 30, SSwithin = 60, and an F-ratio with df = 2, 15. For this analysis, what is the F-ratio?
15/4 = 3.75
A researcher reports a two-sample t-statistic with df = 16. Population variances are assumed to be equal. How many participants were in the entire study?
18
An ANOVA is used to evaluate the mean differences among three treatment conditions with a sample of n = 12 participants in each treatment. For this study, what is df between treatments?
2
An analysis of variances produces dftotal = 29 and dfwithin = 27. For this analysis, what is dfbetween?
2
An independent-measures study with n = 6 in each sample produces a sample mean difference of 4 points and a pooled variance of 12. What is the value for the t statistic?
2
Two samples each have n = 4 scores. If the first sample has a variance of 10 and the second sample has a variance of 6, what is the estimated standard error for the sample mean difference?
2
The critical F value with 6 numerator and 60 denominator degrees of freedom at a = .05 is
2.25
An independent-measures research study compares three treatment conditions using a sample of n = 10 in each treatment. For this study, the three sample means are x¯¯1 = 1, x¯¯2 = 2, and x¯¯3 = 3. For the ANOVA, what value would be obtained for SSbetween?
20
In a two-way ANOVA, there are 5 levels for factor A and 4 levels for factor B, and three observations within each cell. The number of treatments in this experiment will be:
20
The following information was obtained from independent random samples. Assume normally distributed populations with equal variances. Sample Mean 1: 45 2: 42 Sample Variance 1: 85 2: 90 Sample Size 1: 10 2: 12 The degrees of freedom for the t-distribution are
20
SSTR = 6,750 H0: m1 = m2 = m3 = m4 SSE = 8,000 Ha: at least one mean is different n1=n2=n3=n4= 20 The test statistic to test the null hypothesis equals
21.375
SSTR = 6,750 H0: m1 = m2 = m3 = m4 SSE = 8,000 Ha: at least one mean is different n1=n2=n3=n4= 20 The mean square between treatments (MSTR) equals
2250
An analysis of variance produces SSTR = 40 and MSTR = 20. In this analysis, how many treatment conditions are being compared?
3
An analysis of variances produces dftotal = 29 and dfwithin = 27. For this analysis, how many treatment conditions are being compared?
3
The data from an independent-measures research study produce a sample mean difference of 4 points and a pooled variance of 18. If there are n = 4 scores in each sample, what is the estimated standard error for the sample mean difference?
3
An ANOVA procedure is used for data obtained from four populations. Four samples, each comprised of 30 observations, were taken from the four populations. The numerator and denominator (respectively) degrees of freedom for the critical value of F are
3 and 116
An ANOVA procedure is used for data that was obtained from four sample groups each comprised of five observations. The degrees of freedom for the critical value of F are
3 and 16
A researcher reports an F-ratio with df = 2, 18 from an independent-measures research study. Based on the df values, how many treatments were compared in the study, and what was the total number of subjects participating in the study?
3 treatments and 21 subjects
The critical F value with 8 numerator and 29 denominator degrees of freedom at a = 0.01 is
3.20
For an independent-measures ANOVA comparing three treatments with a sample of n = 5 in each treatment, what is the critical value for the F-ratio using a = .05?
3.88
Given the significance level 0.01, the F-value for the degrees of freedom, df = (9,15) is:
3.89
An ANOVA is used to evaluate the mean differences among three treatment conditions with a sample of n = 12 participants in each treatment. For this study, What is dfwithin treatments?
33
In general, what factors are most likely to reject the null hypothesis for an ANOVA?
33
An ANOVA is used to evaluate the mean differences among three treatment conditions with a sample of n = 12 participants in each treatment. For this study, what is dftotal?
35
For an analysis of variance comparing four treatments, MSTR = 12. What is the value of SSTR?
36
Two samples, each with n = 5 scores, have a pooled variance of 40. What is the estimated standard error for the sample mean difference?
4
An ANOVA procedure is used for data obtained from five populations. five samples, each comprised of 20 observations, were taken from the five populations. The numerator and denominator (respectively) degrees of freedom for the critical value of F are
4 and 95
The following information was obtained from independent random samples. Assume normally distributed populations with equal variances. Sample 1 Sample 2 Sample Mean 1: 45 2: 42 Sample Variance 1: 85 2: 90 Sample Size 1: 10 2: 12 The standard error of x¯¯1−x¯¯2 is
4.0
In an analysis of variance problem if SST = 120 and SSTR = 80, then SSE is
40
Consider the following ANOVA table: Source of Variation SS df MS F Treatments 128 4 32 2.963 Error 270 25 10.8 Total 398 29 The number of treatments is:
5
In a one-way ANOVA involving five treatments, 13 observations were recorded for each of the five treatments (a total of 65 observations). The following information is provided. SSTR = 200 (Sum Square Between Treatments) SST = 800 (Total Sum Square) The test statistic is
5
An ANOVA procedure is applied to data obtained from 6 samples where each sample contains 20 observations. The degrees of freedom for the critical value of F are
5 numerator and 114 denominator degrees of freedom
An analysis of variance produces SStotal = 80 and SSwithin = 30. For this analysis, what is SSbetween?
50
In a one-way ANOVA involving five treatments, 13 observations were recorded for each of the five treatments (a total of 65 observations). The following information is provided. SSTR = 200 (Sum Square Between Treatments) SST = 800 (Total Sum Square) The mean square between treatments (MSTR) is
50.00
One-way ANOVA is performed on independent samples taken from three normally distributed populations with equal variances. The following summary statistics were calculated: n1 = 6 x¯1= 50 s1 = 5.2 n2 = 8 x¯2= 55 s2 = 4.9 n3 = 6 x¯3= 51 s3 = 5.4 the grand mean equals:
52.3
The number of degrees of freedom for the denominator in a one-way ANOVA test for 6 population means with 10 observations sampled from each population is:
54
Consider the following partial ANOVA table: Source of Variation treatments SS df MS F Error 102 17 4.533 48.75 3.75 Total 150.75 19 The numerator and denominator degrees of freedom (blank spots) are, respectively,
6 and 13
One-way ANOVA is performed on three independent samples with n1 = 6, n2 = 7, and n3 = 8. The critical value obtained from the F-table for this test at the 1% level of significance equals:
6.01
In a one-way ANOVA involving five treatments, 13 observations were recorded for each of the five treatments (a total of 65 observations). The following information is provided. SSTR = 200 (Sum Square Between Treatments) SST = 800 (Total Sum Square) The number of degrees of freedom corresponding to within treatments is
60
n a one-way ANOVA involving five treatments, 13 observations were recorded for each of the five treatments (a total of 65 observations). The following information is provided. SSTR = 200 (Sum Square Between Treatments) SST = 800 (Total Sum Square) The number of degrees of freedom corresponding to within treatments is
60
One-way ANOVA is applied to independent samples taken from three normally distributed populations with equal variances. The following summary statistics were calculated: n1=12 x¯1= 40 s1 = 5 n2=12 x¯2= 42 s2 = 6 n3=12 x¯3= 50 s3 = 4 The between-treatments variation equals:
672
One-way ANOVA is applied to independent samples taken from three normally distributed populations with equal variances. The following summary statistics were calculated: n1=12 x¯1= 40 s1 = 5 n2=12 x¯2= 42 s2 = 6 n3=12 x¯3= 50 s3 = 4 The between-treatments variation equals:
672
An analysis of variances produces dfbetween = 3 and dfwithin = 24. If each treatment has the same number of participants, then how many participants are in each treatment?
7
Two samples of sizes 35 and 50 are independently drawn from two normal populations, where the unknown population variances are assumed to be equal. The number of degrees of freedom of the equal-variances t-test statistic is:
83
Two samples of sizes 35 and 50 are independently drawn from two normal populations, where the unknown population variances are assumed to be equal. The number of degrees of freedom of the equal-variances t-test statistic is
83.
A research report concludes that there are significant differences among treatments, with "F2, 24 = 8.62, p-value < .01." If the same number of participants was used in all of the treatment conditions, then how many individuals were in each treatment?
9
When the following hypotheses are being tested at a level of significance of a, H0: m1- m2 >= 0 Ha: m1- m2 < 0 the null hypothesis will be rejected if the p-value is
< a
When the following hypotheses are being tested at a level of significance of a, H0: m1- m2 ³ 0 Ha: m1- m2 < 0 the null hypothesis will be rejected if the p-value is
< a
In order to test the following hypotheses at an a level of significance, H0: m1- m2 £ 0 Ha: m1- m2 > 0 the null hypothesis will be rejected if the test statistic Z is
>= Za
In order to test the following hypotheses at an a level of significance, H0: m1- m2 <= 0 Ha: m1- m2 > 0 the null hypothesis will be rejected if the test statistic Z is
>=Za
If other factors are held constant, which of the following sets of data is most likely to produce a significant mean difference?
A sample mean difference of 10 points with n = 10 for both samples
For an ANOVA comparing three treatment conditions, what is stated by the alternative hypothesis (Ha)?
At least one of the three population means is different from another mean
Which of the following research situations would be most likely to use an independent-samples design?
Compare the mathematics skills for 9th-grade boys versus 9th-grade girls
The distribution of the test statistic for analysis of variance is the:
F-distribution
An independent-measures research study uses two samples, each with n = 10 participants. If the data produce a t statistic of t = 2.095, which of the following is the correct decision for a two-tailed hypothesis test? Assume equal population variances
Fail to reject the null hypothesis with either a = .05 or a = .01
An independent-measures research study uses two samples, each with n = 10 participants. If the data produce a t statistic of t = 2.095, which of the following is the correct decision for a two-tailed hypothesis test? Assume equal population variances.
Fail to reject the null hypothesis with either a = .05 or a = .01
If we are interested in testing whether the mean of population 1 is significantly different from the mean of population 2, the correct null hypothesis is
H0: m1- m2 = 0
If we are interested in testing whether the mean of population 1 is significantly larger than the mean of population 2, the correct null hypothesis is
H0: m1- m2 = 0
A political analyst in Texas surveys a random sample of registered Democrats and compares the results with those obtained from a random sample of registered Republicans. This would be an example of
Independent samples
A political analyst in Texas surveys a random sample of registered Democrats and compares the results with those obtained from a random sample of registered Republicans. This would be an example of:
Independent samples.
One sample has n = 10 scores and a variance of s2 = 20, and a second sample has n = 15 scores and a variance of s2 = 30. What can you conclude about the pooled variance for these two samples?
It will be closer to 30 than to 20.
In general, what factors are most likely to reject the null hypothesis for an ANOVA?
Large mean differences and small variances
The F ratio in a one-way ANOVA is the ratio of
MSTR/MSE
In an analysis of variance, which of the following is not true?
MStotal = MSbetween + MSSwithin
Which of the following is not a required assumption for the analysis of variance?
Populations have equal means
Which of the following describes a typical distribution of F-statistics?
Positively skewed with all values greater than or equal to zero
Two samples, each with n = 8, produce an equal-variances t-statistic of t = -2.15. Which of the following decisions is justified for a two-tailed test?
Reject H0 with a = .05 but fail to reject with a = .01
An independent-measures research study uses two samples, each with n = 15 participants. If the data produce a t statistic of t = 2.760, which of the following is the correct decision for a two-tailed hypothesis test? Assume equal population variances
Reject the null hypothesis with a = .05 but fail to reject with a = .01
An independent-measures research study uses two samples, each with n = 15 participants. If the data produce a t statistic of t = 2.760, which of the following is the correct decision for a two-tailed hypothesis test? Assume equal population variances.
Reject the null hypothesis with a = .05 but fail to reject with a = .01
An analysis of variance is used to evaluate the mean differences for a research study comparing four treatments with a separate sample of n = 8 in each treatment. If the data produce an F-ratio of F = 4.60, which of the following is the correct statistical decision?
Reject the null hypothesis with either α = .05 or α= .01
In one-way ANOVA, suppose that there are five treatments with n1 = n2 = n3 = 10, and n4 = n5 = 7. Then the mean square for error, MSE, equals:
SSE / 39
In an analysis of variance, which of the following is determined by the size of the sample mean differences?
SSbetween
One sample has a variance of s2 = 10 and a second sample has a variance of s2 = 6. Which of the following most accurately describes the pooled variance for the two samples?
Somewhere between 6 and 10
In an ANOVA, what is represented by the letter N?
The total number of observations in the research study
What is assumed by the "equal variances" assumption?
The two populations have equal variances.
For an ANOVA comparing three treatment conditions, what is stated by the null hypothesis (H0)?
There are no differences between any of the population means
The equation: SST = SSA + SSB + SSAB + SSE, applies to which ANOVA model?
Two-way ANOVA
A researcher predicts that scores in treatment A will be higher than scores in treatment B. If the mean for the 10 participants in treatment A is 4 points higher than the mean for the 10 participants in treatment B and the data produce t = 2.095, what decision should be made using the equal variances approach?
With a = .05, reject H0 for a one-tailed test but fail to reject for a two-tailed test
If we are interested in testing whether the mean of items in population 1 is larger than the mean of items in population 2, the
alternative hypothesis should state m1 - m2 > 0
If we are interested in testing whether the mean of items in population 1 is significantly smaller than the mean of items in population 2, the
alternative hypothesis should state m1- m2 < 0
A balanced experiment requires that:
an equal number of persons or test units receives each treatment.
If two independent large samples are taken from two populations, the sampling distribution of the difference between the two sample means
can be approximated by a normal distribution
A researcher uses analysis of variance to test for mean differences among three treatments with a sample of n = 12 in each treatment. The F-ratio for this analysis would have what df values?
df = 2, 33
In order to determine whether or not the means of two populations are equal,
either a t test or an analysis of variance can be performed
One sample of n = 8 scores has a variance of s2 = 6 and a second sample of n = 8 scores has s2 = 10. If the pooled variance is computed for these two samples, then the value obtained will be ______.
exactly halfway between 6 and 10
A term that means the same as the term "variable" in an ANOVA procedure is
factor
One-way ANOVA is applied to three independent samples having means 8, 15, and 20, respectively. If each observation in the third sample were increased by 20, the value of the F-statistic would:
increase
To test whether or not there is a difference between treatments A, B, and C, a sample of 12 observations has been randomly assigned to the 3 treatments. You are given the results below. Treatment: A B C Observation A 20 30 25 33 B 22 26 20 28 C 40 30 28 22 The null hypothesis for this ANOVA problem is
m1=m2=m3
A researcher predicts that scores in treatment A will be higher than scores in treatment B. Which of the following is the correct null hypothesis for a one-tailed test evaluating this prediction?
mA <= mB
A researcher reports an independent-measures t-statistic with df = 30. If the two samples are the same size (n1 = n2), and population variances are equal, then how many individuals are in each sample?
n = 16
Independent simple random samples are taken to test the difference between the means of two populations whose variances are known. The sample sizes are n1 = 38 and n2 = 42. The correct distribution to use is the
normal distribution
In testing the difference between two population means using two independent samples, the population standard deviations are assumed to be unknown, each sample size is ≥ 30, and the calculated test statistic z = 2.56. If the test is two-tail and 1% level of significance has been specified, the conclusion should be to:
not to reject the null hypothesis
A survey will be conducted to compare the United Way contributions made by sales people from three county retail corporations. Sales people are to be randomly selected from each of the three corporations and the dollar amounts of their contribution recorded. The ANOVA model most likely to fit this situation is the:
one-way analysis of variance.
Two samples of ten each from the male and female workers of a large company have been taken. The data involved the wage rate of each worker. To test whether there is any difference in the average wage rate between male and female workers a pooled-variances t-test will be considered. Another test option to consider is ANOVA. The most likely ANOVA to fit this test situation is the:
one-way analysis of variance.
When the p-value is used for hypothesis testing, the null hypothesis is rejected if
p-value < a
For testing the following hypothesis at 95% confidence, the null hypothesis will be rejected if H0: m1 - m2 <= 0 Ha: m1 - m2 > 0
p-value <= 0.05
In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference x¯1−x¯2 if the:
populations are at least normally distributed with equal variances.
Salary information regarding male and female employees of a large company is shown below. Male Female Sample Size 64 36 Sample Mean Salary (in $1,000) 44 41 Population Variance (σ2) 128 72 At 95% confidence, the conclusion is the
salaries of males and females are equal
Salary information regarding male and female employees of a large company is shown below. Male Female Sample Size Male: 64 Female: 36 Sample Mean Salary (in $1,000) Male: 44 Female: 41 Population Variance (σ2) Male: 128 Female: 72 At 95% confidence, the conclusion is the
salaries of males and females are equal
The required condition for using an ANOVA procedure on data from several populations is that the
sampled populations have equal variances
Independent simple random samples are taken to test the difference between the means of two populations whose standard deviations are not known, but are assumed to be equal. The sample sizes are n1 = 25 and n2 = 35. The correct distribution to use is the
t distribution with 58 degrees of freedom
Independent simple random samples are taken to test the difference between the means of two populations whose variances are not known, but are assumed to be equal. The sample sizes are n1 = 32 and n2 = 40. The correct distribution to use is the
t distribution with 70 degrees of freedom
In hypothesis testing if the null hypothesis is rejected
the alternative hypothesis is true
In hypothesis testing if the null hypothesis is rejected,
the alternative hypothesis is true
In the analysis of variance procedure (ANOVA), "factor" refers to
the independent variable
Which of the following does not need to be known in order to compute the p-value?
the level of significance
The ANOVA procedure is a statistical approach for determining whether or not
the means of two or more populations are equal
The one-way ANOVA model assumes that each individual observation is considered to be the sum of the overall population mean for all the treatments plus the effect of the treatment plus
the random error associated with the sampling process
When the effect of a level for one factor depends on which level of the other factor is present, use:
two-way analysis of variance
The variation that reflects the effect of the factor levels is known as the
variation between the groups
The F-statistic in a one-way ANOVA represents the:
variation between the treatments divided by the variation within the treatments.
The variation that reflects the random error from the sampling process is known as the
variation within the groups
If a hypothesis is rejected at 95% confidence, it
will also be rejected at 90% confidence