Question

12. Comparing ANOVA and the t test for an independent-measures hypothesis test

Suzanne Zeedyk, a developmental psychologist at Dundee University’s School of Psychology, conducted a pilot study in which parents started a half-hour walk with their infants in a parent-facing or an away-facing stroller and then switched to the other type of stroller midway. Her results suggest that parents talked less to the babies, the babies had higher heart rates, and they were less likely to fall asleep in away-facing strollers than in parent-facing strollers.

You are interested in testing the hypothesis that babies who travel in parent-facing strollers have different expressive vocabularies than babies who travel in away-facing strollers. You randomly assign 12 newborns to parent-facing strollers and 12 newborns to away-facing strollers. You then test the babies’ expressive vocabularies at age 36 months using the Expressive Vocabulary Test (EVT), which is designed primarily to assess children’s expressive vocabulary.

The sample means and sums of squares of the scores for each of the groups are presented in the following table.

Group	Sample Mean	Sum of Squares
Away-facing	98.1	2,064.59
Parent-facing	106.2	1,404.59

You decide to use an ANOVA at α = 0.01 to test the null hypothesis that there is no difference between the groups. The ANOVA table follows. Calculate the F-ratio and enter it into the table.

Source of Variation	ANOVA Table
	Sum of Squares	Degrees of Freedom	Mean Square	F
Between Treatments	393.66	1	393.66
Within Treatments	3,469.18	22	157.69
Total	3,862.84	23

Use the Distributions tool that follows to find Fcriticalcritical at a significance level of α = 0.01, the value of F that bounds the critical region for the F-ratio test statistic.

F Distribution

Numerator Degrees of Freedom = 26

Denominator Degrees of Freedom = 26

0.001.002.003.004.005.006.007.008.00F.5000.50001.000

Fcriticalcritical at α = 0.01 is   .

Now evaluate the null hypothesis that the population means for all treatments are equal. At a significance level of α = 0.01, the null hypothesis is   . You find that you   conclude that the direction that the stroller faces influences a child’s expressive vocabulary at 36 months.

Now you decide to use a t test to test the hypothesis that there is no difference between the groups. The estimated standard error (sM1 – M2M1 – M2) is 5.13, so the t test statistic is   .

Use the following tool to find the critical regions for α = 0.01.

t Distribution

Degrees of Freedom = 21

-3.0-2.0-1.00.01.02.03.0t.2500.5000.2500.5000.5000-0.6860.6860.000

The critical t-scores (the values for t-scores that separate the tails from the main body of the distribution, forming the critical regions) are   .

Now use the tool to evaluate the null hypothesis. At a significance level of α = 0.01, the null hypothesis is   . You find that you   conclude that the direction the stroller faces influences a child’s expressive vocabulary at 36 months.

When you evaluated the mean difference from the independent-measures study comparing only two samples, the ANOVA and the t test resulted in   conclusion.

Answer 1

the value of F stat is 393.66/157.69 = 2.496

Fcritical at α = 0.01 is 7.95

Now evaluate the null hypothesis that the population means for all treatments are equal. At a significance level of α = 0.01, the null hypothesis is not rejected . you do not conclude that the direction that the stroller faces influences a child’s expressive vocabulary at 36 months.

Now you decide to use a t test to test the hypothesis that there is no difference between the groups. The estimated standard error (sM1 – M2M1 – M2) is 5.13, so the t test statistic is   .-1.58

The critical t-scores (the values for t-scores that separate the tails from the main body of the distribution, forming the critical regions) are +/-2.82

Now use the tool to evaluate the null hypothesis. At a significance level of α = 0.01, the null hypothesis is not rejected . you cannot conclude that the direction the stroller faces influences a child’s expressive vocabulary at 36 months.

When you evaluated the mean difference from the independent-measures study comparing only two samples, the ANOVA and the t test resulted in same conclusion.