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 receptive vocabularies than babies who travel in away-facing strollers. You randomly assign 18 newborns to parent-facing strollers and 18 newborns to away-facing strollers. You then test the babies’ receptive vocabularies at age 18 months using the Peabody Picture Vocabulary Test—Revised (PPVT-R), which is designed primarily to assess children’s receptive 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 92.7 2,828.97 Parent-facing 104.3 2,530.28 You decide to use an ANOVA at α = 0.05 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 1,211.04 1 1,211.04 Within Treatments 5,359.25 34 157.63 Total 6,570.29 35 Use the Distributions tool that follows to find Fcriticalcritical at a significance level of α = 0.05, the value of F that bounds the critical region for the F-ratio test statistic. (Note: Do not use the t-tables. In some cases, when the degrees of freedom are not exact, they do not provide answers with enough precision for this exercise.) 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 = 0.05 is . Now evaluate the null hypothesis that the population means for all treatments are equal. At a significance level of α = 0.05, the null hypothesis is . You find that you conclude that the direction that the stroller faces influences a child’s receptive vocabulary at 18 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 4.19, so the t test statistic is . Use the following tool to find the critical regions for α = 0.05. 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 = 0.05, the null hypothesis is . You find that you conclude that the direction the stroller faces influences a child’s receptive vocabulary at 18 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.

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 receptive vocabularies than babies who travel in away-facing strollers. You randomly assign 18 newborns to parent-facing strollers and 18 newborns to away-facing strollers. You then test the babies’ receptive vocabularies at age 18 months using the Peabody Picture Vocabulary Test—Revised (PPVT-R), which is designed primarily to assess children’s receptive 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 92.7 2,828.97 Parent-facing 104.3 2,530.28 You decide to use an ANOVA at α = 0.05 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 1,211.04 1 1,211.04 Within Treatments 5,359.25 34 157.63 Total 6,570.29 35 Use the Distributions tool that follows to find Fcriticalcritical at a significance level of α = 0.05, the value of F that bounds the critical region for the F-ratio test statistic. (Note: Do not use the t-tables. In some cases, when the degrees of freedom are not exact, they do not provide answers with enough precision for this exercise.) 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 = 0.05 is . Now evaluate the null hypothesis that the population means for all treatments are equal. At a significance level of α = 0.05, the null hypothesis is . You find that you conclude that the direction that the stroller faces influences a child’s receptive vocabulary at 18 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 4.19, so the t test statistic is . Use the following tool to find the critical regions for α = 0.05. 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 = 0.05, the null hypothesis is . You find that you conclude that the direction the stroller faces influences a child’s receptive vocabulary at 18 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.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.6: Summarizing Categorical Data

Problem 27PPS

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You are interested in testing the hypothesis that babies who travel in parent-facing strollers have different receptive vocabularies than babies who travel in away-facing strollers. You randomly assign 18 newborns to parent-facing strollers and 18 newborns to away-facing strollers. You then test the babies’ receptive vocabularies at age 18 months using the Peabody Picture Vocabulary Test—Revised (PPVT-R), which is designed primarily to assess children’s receptive 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	92.7	2,828.97
Parent-facing	104.3	2,530.28

You decide to use an ANOVA at α = 0.05 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	1,211.04	1	1,211.04
Within Treatments	5,359.25	34	157.63
Total	6,570.29	35

Use the Distributions tool that follows to find Fcriticalcritical at a significance level of α = 0.05, the value of F that bounds the critical region for the F-ratio test statistic. (Note: Do not use the t-tables. In some cases, when the degrees of freedom are not exact, they do not provide answers with enough precision for this exercise.)

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 = 0.05 is .

Now evaluate the null hypothesis that the population means for all treatments are equal. At a significance level of α = 0.05, the null hypothesis is . You find that you conclude that the direction that the stroller faces influences a child’s receptive vocabulary at 18 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 4.19, so the t test statistic is .

Use the following tool to find the critical regions for α = 0.05.

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 = 0.05, the null hypothesis is . You find that you conclude that the direction the stroller faces influences a child’s receptive vocabulary at 18 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.

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