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WorksheetsThe Independent t Test and the Mann-Whitney U-Test
Total questions: 109
Worksheet time: 55mins
Which situation best fits an independent samples t-test?
Testing association between two nominal variables
Comparing mean scores of two unrelated groups
Comparing medians of matched pair groups
Modeling a time trend across repeated measures
Estimating a single population mean only
What does the Mann-Whitney U-test primarily compare between two groups?
Equality of group means exclusively
Correlation between two variables
Overall distribution of values between groups
Proportion differences across categories
Equality of population variances only
In these tests, what is the variable that defines the groups called?
Confounding variable influencing both
Nuisance variable with no impact
Outcome variable measuring effect
Grouping variable defining categories
Control variable held constant
Which assumption is required for the independent samples t-test to be valid?
Samples are drawn with replacement
Dependent variable is approximately normal
Groups share identical sample sizes
Outliers are removed by default
Independent variable is continuous only
For the t-test, how many categories should the grouping variable have?
A single continuous scale of scores
Three or more ordered categories
Any number of categories is acceptable
Exactly two mutually exclusive categories
Two overlapping non-exclusive groups
Which statement best distinguishes the two tests?
t-test compares means; U-test compares distributions
Both tests compare medians exclusively
t-test requires ordinal data; U-test requires normal data
Both tests require equal variances always
t-test is nonparametric; U-test is parametric
Who originally published the independent t-test under the pseudonym “Student”?
D. Ransom Whitney, a graduate
Karl Pearson, a biologist
Henry B. Mann, an economist
Ronald Aylmer Fisher, a statistician
William Sealy Gosset, an Englishman
Which research question most appropriately uses the Mann-Whitney U-test?
Estimating the mean BMI in a single city sample
Comparing average heights of unrelated adults with normals
Testing matched pre-post exam score differences
Evaluating correlation between age and income
Comparing delivery times by drug type with skewed times
Under the null hypothesis for these two-group tests, what is assumed?
Group variances are exactly equal in samples
Sample means are identical to medians
Effect sizes are at least medium magnitude
No true difference exists between populations
Both groups follow identical normality
Which example describes independent samples?
Patients measured at multiple visits
Workers measured across four weeks
Men and women measured once each
Students measured before and after
Paired twins measured simultaneously
In the heart failure study, what was the grouping variable?
Age category of the participants
Health-related quality-of-life score
Presence of sleeping difficulty status
Hospital site of recruitment
Type of heart failure medication
In the heart failure study, what level is the quality-of-life outcome variable described as?
Nominal-level categories unordered
Ratio-scale exact time to event
Ordinal-level scores treated as continuous
Interval scale with equal negatives
Binary outcome measured as counts
Which condition reflects independence between the two groups for a t-test?
Repeated measures on the same subjects
Both groups sampled from the same units
Participants appear in both group samples
Values from one group do not affect the other
Groups intentionally matched on covariates
When assumptions of the t-test are moderately violated but sample size is large, which is reasonable?
Abandon statistics and use anecdotes
Transform outcome into categories
Always switch to chi-square test
Use paired t-test regardless of design
Proceed with t-test cautiously using robustness
Which statement about the historical development is accurate?
Mann-Whitney U was introduced in 1947
Whitney developed the t-distribution
Student’s t was derived in 2001
U-test originated for beer quality control
Fisher first proposed the U-statistic
Which condition favors using the independent samples t-test over the Mann–Whitney U-test?
Outcome is normally distributed and continuous
Grouping variable has more than two categories
Total sample size is at least eight only
Outcome is nominal with multiple categories
When is the Mann–Whitney U-test an appropriate alternative to the independent t-test?
Outcome is nominal with unequal groups
Outcome is ratio and paired within subjects
Outcome is binary with paired observations
Outcome is at least ordinal with independent samples
In the weight-loss study, what is the grouping variable and how many levels does it have?
Gender with two levels
Clinic site with two levels
Diet type with three levels
Baseline BMI with four levels
Which statement correctly specifies the null and alternative hypotheses for comparing mean weight loss between men and women?
H0: μmen < μwomen; HA: μmen > μwomen
H0: μmen ≠ μwomen; HA: μmen = μwomen
H0: μmen > μwomen; HA: μmen = μwomen
H0: μmen = μwomen; HA: μmen > μwomen
A two-tailed independent samples t-test with n = 32 uses what degrees of freedom?
30 degrees of freedom
31 degrees of freedom
32 degrees of freedom
29 degrees of freedom
For α = .05 and df = 30 in a two-tailed t-test, which critical value is used to judge significance?
t* ≈ 1.96
t* ≈ 1.645
t* ≈ 2.576
t* ≈ 2.042
Which assumption is required for the independent samples t-test in this context?
Matched pairs within the same subjects
Independent random samples from each group
Outcome measured on a nominal scale
Extreme skewness in both groups
Why might one choose a two-tailed test even with a directional hypothesis (men higher than women)?
Two-tailed tests eliminate the need for p-values
Two-tailed tests always provide more power than one-tailed
Directional tests are not defined for t-statistics
Standard practice and software defaults favor two-tailed tests
Which statement correctly specifies the null and alternative hypotheses for comparing mean weight loss between two independent groups (men vs. women)?
H0: μmen > μwomen; HA: μmen < μwomen
H0: μmen < μwomen; HA: μmen > μwomen
H0: μmen ≠ μwomen; HA: μmen = μwomen
H0: μmen = μwomen; HA: μmen ≠ μwomen
Given sample variances s12=29.84 (women, n1=15 ) and s22=36.26 (men, n2=17 ), which F statistic is used to test homogeneity of variance?
F = 36.26 ÷ 29.84 = 1.22
F = (36.26 − 29.84) ÷ 2
F = 29.84 ÷ 36.26 = 0.82
F = (36.26 + 29.84) ÷ 2
Using pooled-variance t for two independent samples, which expression gives the standard error term multiplied by (1/n1 + 1/n2)?
SEpooled = sqrt[sp2(1/n1+1/n2)]
SEpooled = s1/√n1 + s2/√n2
SEpooled=n1+n2s12+s22
SEpooled = (x̄1 − x̄2)/t
With x̄1 = 12.13 (women), x̄2 = 18.59 (men), n1 = 15, n2 = 17, and pooled SE ≈ 2.047, what is the computed t-statistic?
t ≈ (18.59 − 12.13)/6.55 ≈ 0.99
t ≈ (12.13 − 18.59)/6.55 ≈ −0.99
t ≈ (18.59 − 12.13)/2.047 ≈ 3.16
t ≈ (12.13 − 18.59)/2.047 ≈ −3.16
For α = 0.05 with df = n1 + n2 − 2 = 30, which decision rule is correct for a two-tailed test?
Reject H0 if t < −1.96
Reject H0 if |t| > 2.042
Reject H0 if t > 1.96
Reject H0 if |t| < 2.042
Using the 95% confidence interval formula (x̄2 − x̄1) ± tcrit(SEpooled), with x̄2 − x̄1 = 6.46, tcrit ≈ 2.042, and SEpooled ≈ 2.047, which CI is correct?
(1.96, 6.46)
(4.41, 8.51)
(2.28, 10.64)
(−3.16, 3.16)
In SPSS for an independent t-test, how should a binary grouping variable like gender typically be coded before analysis?
Use text labels female, male
Enter decimals 0.5 and 1.5
Assign numbers 1 and 2
Leave cells blank for males
Which SPSS menu sequence opens the independent-samples t-test dialog?
Analyze > Descriptive Statistics > Explore
Transform > Compute > Independent-Samples Test
Analyze > Nonparametric Tests > Two Independent Samples
Analyze > Compare Means > Independent-Samples T Test
Where should the outcome variable (e.g., weight loss) be placed in the independent-samples t-test dialog?
Grouping Variable box
Test Variables box
Filter Cases box
Options submenu only
After selecting the grouping variable, which action specifies the two groups for the test in SPSS?
Click Recode into Different Variables
Click Define Groups and enter codes
Click Split File and select compare groups
Click Paste and run syntax
From the provided SPSS output, Levene’s test for equality of variances has p = .078. Which row of the t-test should be interpreted?
Equal variances assumed row
Neither row; test is invalid
Equal variances not assumed row
Both rows averaged together
The t statistic for the equal variances assumed row is approximately −3.159 with df = 30 and p ≈ .004. What is the correct decision at α = .05?
Conclude unequal variances
Reject the null; means differ
Fail to reject; means are similar
Increase sample size only
Given group means of about 12.13 for group 1 (n=15) and 18.88 for group 2 (n=17), which interpretation aligns with the sign of t (negative)?
The sign is unrelated to group means
Group 2 mean is larger than group 1
Mean of group 1 is larger than group 2
Both group means are equal in size
Which assumption is required for the independent samples t-test when comparing two groups?
Each variable is measured on a nominal scale
Sample sizes are always identical
Population medians are exactly equal
Observations within groups are independent
When group variances appear equal, which formula should be used for the independent samples t-test?
Paired-samples t-test formula for matched pairs
Nonparametric Mann–Whitney U procedure
Welch-adjusted test for unequal variances
Pooled variances formula for equal variances
The F_max test for homogeneity of variance compares which quantities?
Larger mean divided by smaller mean
Sum of variances across both groups
Difference between standard deviations
Larger variance divided by smaller variance
Suppose F_max is below the critical value. What is the proper decision about variances?
Transform data to meet all assumptions
Abandon t-tests and use medians
Conclude heterogeneity and use separate test
Assume homogeneity and use pooled test
Levene’s test has which practical characteristic compared with F_max?
Less sensitive to normality violations
Identical computation to the Hartley test
Only valid for perfectly normal data
More dependent on exact sample equality
For equal variances, the pooled standard error used in the t-statistic depends on which feature?
The difference between group variances
A pooled variance from both groups
Only the larger group’s variance
Only the smaller group’s variance
Degrees of freedom for the equal-variances independent t-test with n1 and n2 are computed as:
n1 + n2 − 2
min(n1,n2) − 1
(n1 − 1) + (n2 − 1) + 2
n1 + n2 − 1
A 95% confidence interval for the mean difference (x̄1 − x̄2) using equal variances is computed as:
Pooled mean × t_crit ÷ difference
Group standard error − t_crit × SD
Point estimate ± z_crit × sample SD
Point estimate ± t_crit × SE_pooled
If a 95% confidence interval for (μ1 − μ2) is entirely above 0, what conclusion is justified at α = .05 for a two-tailed test?
There is no significant difference between groups
The groups differ significantly in the positive direction
Variances must be unequal in the population
The null is accepted with small error
In SPSS output, Levene’s test p-value greater than .05 suggests which t-test row should be interpreted?
Equal variances assumed row
Equal variances not assumed row
Nonparametric tests summary
Paired t-test results section
A computed t-statistic with absolute value larger than the critical value implies which decision?
Retain the null hypothesis of equal means
Reject the null hypothesis of equal means
Report only effect size without p-value
Increase the significance level to .10
When F_max exceeds its critical value, the appropriate action for independent two-sample comparison is to:
Use separate-variances t-test (Welch’s)
Ignore variance inequality and proceed
Use pooled-variances t-test regardless
Switch to paired-samples approach
What does the independent samples t-test primarily test about two group means?
Whether their difference equals zero on average
Whether both groups are normally distributed
Whether both groups have identical variances
Whether sample medians are exactly the same
Which statement best describes the t-distribution used for the t-test?
Symmetric, bell-shaped, mean equal to zero
Skewed right with heavy tails
Flat with uniform probabilities
Bimodal with two distinct peaks
In hypothesis testing with a two-tailed independent t-test at alpha = .05, what defines the rejection regions?
Only the upper tail beyond +tcrit
Only the lower tail beyond −tcrit
Middle 95 percent around zero
Critical t values at both tails
How does increased within-group variability affect the t statistic, holding mean difference and n constant?
Decreases t by increasing error
Makes t negative automatically
Increases t by reducing error
Leaves t unchanged entirely
Which change most likely increases power for detecting a true mean difference?
Higher alpha to lower .01
Greater measurement noise
Larger sample size per group
More unequal group variances
A study compares two groups with the same mean difference. Which scenario is more likely to yield significance?
Unequal sample sizes but same SDs
Smaller standard deviations in both groups
Extremely skewed score distributions
Larger standard deviations in both groups
When should a one-tailed t-test be planned instead of a two-tailed test?
When normality cannot be assessed reliably
When results are already statistically significant
When a directional effect is theoretically justified
When variances are exactly equal by design
What does a 95% confidence interval for the mean difference represent conceptually?
A range likely to contain the true difference
The most probable sample mean values
The interval where t equals exactly zero
The bounds that guarantee significance
Cohen’s d for an independent t-test is defined as which ratio?
Variance difference divided by sample size
Sample mean divided by population mean
P-value divided by alpha level
Mean difference divided by standard deviation
Power of a t-test is best defined as which probability?
Failing to reject a true null hypothesis
Rejecting a true null hypothesis
Obtaining a p-value exactly equal to alpha
Rejecting a false null hypothesis
A power table for a two-tailed t-test at α = .05 shows required sample sizes per group for different standardized effect sizes d and desired power. Which statement best describes how required sample size changes with effect size and power?
Larger d and lower power require larger samples
Larger d and higher power require larger samples
Smaller d and higher power require smaller samples
Smaller d and higher power require larger samples
You plan a two-tailed t-test at α = .05 comparing two independent groups. You want power ≈ .80 to detect a moderate effect size around d = .50. Using standard power tables for the t-test, which per-group sample size is closest to what you need?
About 64 per group
About 50 per group
About 33 per group
About 17 per group
Which statement best describes the Mann-Whitney U-test?
Nonparametric test comparing two independent groups
Parametric test comparing paired observations
Parametric test estimating population variance
Nonparametric test for more than two dependent groups
Which assumption is most appropriate for using the Mann-Whitney U-test in comparing two groups?
Outcome is normally distributed
Sample size exceeds one hundred
Outcome is at least ordinal scale
Groups are matched pairs only
In the pain relief study, what is the independent variable?
Type of pain control used
Time since injury days
Self-rated pain relief score
Patient age in years
What is the null hypothesis for the Mann-Whitney U-test in the pain relief example?
Means of pain relief are both larger
Painkillers produce higher median relief
TENS produces higher mean pain relief
No difference in distributions of pain relief
Given n1 = 11 for TENS and n2 = 12 for painkillers with alpha = .05 two-tailed, which criterion determines significance for U?
Computed U must be larger than critical value
Computed U must equal degrees of freedom
Computed U must exceed sample medians
Computed U must be less than critical value
Which descriptive statistics are recommended before computing the Mann-Whitney U-test?
Median and interquartile range for each group
Mean and standard deviation only
Mode and range for pooled data
Skewness and kurtosis for both groups
From the example, which pair of summaries matches the groups’ medians and IQRs?
TENS median 6.5 IQR 7; Painkillers median 16 IQR 2
TENS median 16 IQR 2; Painkillers median 6.5 IQR 7
TENS median 16 IQR 7; Painkillers median 13 IQR 2
TENS median 13 IQR 10; Painkillers median 16 IQR 2
Which step occurs first when computing the Mann-Whitney U-statistic from raw scores?
Pool variances and compute t-statistic
Transform scores to z-scores before ranking
Subtract group means and square differences
Rank all observations from smallest to largest
When ranking data for a Mann–Whitney U-test and two observations are tied for ranks 2 and 3, what rank should each tied observation receive?
Both receive rank 2.5
Both receive rank 3
Both receive rank 2
Assign ranks 2 and 3 randomly
A dataset includes five identical scores that would occupy ranks 17 through 21 if distinct. What is the correct average rank assigned to each of these tied scores?
19.5
19
18
20
An SPSS Mann–Whitney U-test reports an asymptotic significance of .000 at alpha = .05 for testing whether two groups have the same distribution of a variable. Which decision is most appropriate?
Fail to reject the null hypothesis
Reject the null hypothesis
Conclude groups are identical in medians
Increase alpha to .10 first
Which SPSS menu path initiates the Mann–Whitney test for two independent samples?
Graphs > Legacy Dialogs > Scatter/Dot
Transform > Compute Variable > Ranks
Analyze > Compare Means > Independent-Samples T Test
Analyze > Nonparametric Tests > Independent Samples
In preparing variables for the Mann–Whitney test in SPSS, how should the group and outcome variables be measured?
Group nominal, outcome scale
Group scale, outcome nominal
Both variables scale
Both variables nominal
After choosing Independent Samples under Nonparametric Tests, which action correctly assigns variables?
Move outcome into Test Fields and group into Groups
Move group into Test Fields and outcome into Groups
Leave fields empty to use defaults
Move both variables into Test Fields only
When customizing tests in the dialog, which specific test should be selected to compute the Mann–Whitney statistic for two groups?
Kruskal–Wallis H (k samples)
Median test for k samples
Mann–Whitney U (2 samples)
Wilcoxon Signed-Rank test
Two groups have sizes n = 11 and m = 12 with the smaller group’s rank sum R_n = 190. Using U = R_n − n(n+1)/2, what is U?
124
66
132
86
Given n = 11, m = 12, and U = 124, compute U' using U' = n×m − U.
12
8
20
24
For small samples (both group sizes fewer than 20), which value is compared to critical tables to determine significance?
The larger of U and U'
Only U regardless of size
The smaller of U and U'
Only U' regardless of size
When determining significance with the Mann–Whitney U-test for small samples, what is compared to a tabled critical value to decide whether to reject the null hypothesis?
The larger of U and U′ values
The average of U and U′ values
The sum of U and U′ values
The smaller of U and U′ values
If both groups each have more than 20 observations, which approach is used to assess Mann–Whitney significance instead of relying on small-sample U tables?
Transform ranks to percentiles
Switch to a paired t-test
Look up a larger U table
Compute a z-score for U
In a study comparing TENS users to painkiller users, the computed smaller U was 8 and the critical value was 34. Which conclusion is justified at α = 0.05?
Reject the null; painkillers are superior
Fail to reject; groups are similar
Reject the null; TENS shows greater relief
Inconclusive due to tied ranks
Conceptually, why does a very small U indicate a statistically significant difference between two groups in the Mann–Whitney test?
Medians must be identical across groups
Sample sizes are exactly equal
Ranks are evenly split across groups
Ranks cluster in one group more
The independent samples t test is best described as
none of the above
a nonparametric test
a parametric test
a type of Mann-Whitney U-test
The independent samples t test is used to determine differences in the means of
any number of groups
three groups only
four groups only
two groups only
If the variances of the two groups being compared are significantly different, which independent samples t test should be used?
The Mann-Whitney U-test
The independent samples t test for pooled variances
The independent samples t test for separate variances
Either test so long as the sample size is over 30 cases
Student’s independent samples t test is best used when the measurement scale of the characteristic of interest is
nominal
ordinal
interval or ratio
all of the above
Student’s independent samples t test is best used when
the grouping variable is dichotomous
a and b only
the total sample size is at least 30
the data are paired
Consider the question: Do women make more visits to their primary care physician in a year than men? Which variable is the grouping variable?
Neither
Number of visits to the physician
Gender
Both
Consider the question: Do people who exercise three times a week or more have lower systolic blood pressure than people who exercise less than three times a week? Which variable should be normally distributed?
Both
Systolic blood pressure
Frequency of exercise
Either one
Consider the question: Do children who are immunized against chickenpox miss fewer days of school than children who are not immunized? Which is the grouping variable?
Neither one
Either one
Immunization status
Number of days of school missed
Consider the question: Do people without health insurance spend more money a year on over-the-counter drugs than people who have some form of health insurance? How many levels (possible values) does the variable “health insurance” have?
Can’t tell
Two
One
Three
Among people with chronic back pain, do those whose back pain is treated by a chiropractor miss fewer days of work than those treated by a primary care physician? Is it possible to use an independent t test if the variable “number of days missed” is recoded into three-category variables (none, less than 3, more than or equal to 3)?
Only if at least 60 people are in the study
Yes
No
Only if an independent random sample is available
The Mann-Whitney U-test is best described as
a special type of independent t test
a parametric test
a nonparametric test
none of the above
The Mann-Whitney U-test is used to determine differences in the distribution of a variable in
two groups only
three groups only
four groups only
any number of groups
If data are normally distributed in only one group, then which of the following tests can be used?
Mann-Whitney U-test
Independent t test for separate variances
Independent t test for pooled variances
Either test as long as the sample size is over 30 cases
The Mann-Whitney U-test is best used when the measurement scale of the characteristic of interest is
interval or ratio
ordinal
b and c
nominal
The Mann-Whitney U-test is best used when
the data are paired
the grouping variable is dichotomous
a and b only are true
the total sample size is at least 8
Consider the question: Do people who eat after 8 pm sleep fewer hours than people who do not eat after 8 pm? Which variable is the grouping variable?
Number of hours of sleep
Eating before or after 8 pm
Neither
Both
Consider the question: Do people who exercise at least three times a week have lower systolic blood pressure than those who exercise less than three times a week? Which variable should be normally distributed to perform a Mann-Whitney U-test?
Neither, because a normal distribution is not an assumption
Both
Systolic blood pressure
Frequency of exercise
Consider the question: Do children immunized against chickenpox miss fewer days of school than those not immunized? Which of the following variables should be dichotomous to perform a Mann-Whitney U-test?
Neither, because a normal distribution is not an assumption
Both
Immunization status
Number of days of school missed
Consider the question: Is there a relationship between insurance status and the number of days absent from work? To use an independent t test, how many possible values should the variable “health insurance” have?
Can’t tell from the information given
Three
Two
One
When comparing two independent groups on a normally distributed quantitative outcome with equal variances, which test is most appropriate?
Paired-samples t-test two-tailed
Wilcoxon signed-rank test
Independent t-test two-tailed
Chi-square test of independence
A researcher recodes “number of days missed” into two categories: none vs one or more. Which test best compares two independent treatment groups on this recoded variable?
Independent samples t-test
Paired-samples t-test
Mann-Whitney U-test
Chi-square test of independence
For a two-tailed independent t-test with alpha = 0.05 and sample sizes n1 = 12 and n2 = 12, what degrees of freedom are used?
24 total degrees
11 total degrees
22 total degrees
23 total degrees
Which statement correctly describes the Mann-Whitney U-test?
Compares medians using paired scores
Requires equal variances across groups
Nonparametric test for two independent groups
Assumes outcome is normally distributed
You plan a two-tailed independent t-test at alpha = .01 with n1 = 25 and n2 = 25. Which step comes first?
Find observed t-statistic value
State null and alternative hypotheses
Compute pooled variance estimate
Compare p-value to alpha level
Given sleep hours for two independent groups (herbal formula vs placebo), which statistic directly tests mean difference under normality?
Pearson correlation coefficient
Independent samples t-statistic
One-sample z-statistic
Wilcoxon signed-rank statistic
A study records number of aggressive acts for children in home care vs day care with skewed counts and outliers. Which analysis is most defensible?
Independent t-test on raw counts
Mann-Whitney U-test on ranks
Paired t-test after matching
One-way ANOVA with two groups
An urban study compares number of physician visits between those with private insurance and those with none; data are non-normal with many zeros. Which is the best primary hypothesis for a Mann-Whitney U-test?
Variances are equal across groups
Proportions of any visit are equal
Means are equal across groups
Median visits equal across groups
