(21.30) Do education programs for preschool children that follow the Montessori

Question

(21.30) Do education programs for preschool children that follow the Montessori method perform better than other programs? A study compared 5-year-old children in Milwaukee, Wisconsin, who had been enrolled in preschool programs from the age of 3.

Step 1: Explain why comparing children whose parents chose a Montessori school with children of other parents would not show whether Montessori schools perform better than other programs.

a. Because the subjects are not an SRS from all the children in the study.

b. This comparison is legitimate, because there is no violation of the conditions for inference.

c. Because the distributions of scores in the two groups are not similar.

d. Because the scores of the two populations are skewed, which violates the conditions for inference.

Step 2: In all, 54 children were assigned to the Montessori school and 112 to other schools at age 3. When the children were 5, parents of 30 of the Montessori children and 25 of the others could be located and agreed to participate in testing. This information reveals a possible source of bias in the comparison of outcomes. Explain why.

a. Because participation in the testing was according to parents agreement.

b. Because the percent of children in the two groups that were compared in the testing differ too much.

c. Because not all the children in each group participated in the testing.

d. Because the number of children assigned to the Montessori school is less than half of the numbers of children at other schools.

Step 3: One of the many response variables was score on a test of ability to apply basic mathematics to solve problems. Here are summaries for the children who took this test: Is there evidence of a difference in the population mean scores? (The researchers used two-sided alternative hypotheses.) Give the two sample test statistic. Give your answer to 3 decimal places

Step 4: What is the P-value for the test?

a. 0.05 < P < 0.1

b. 0.01 < P < 0.05

c. 0.025 < P < 0.05

d. 0.005 < P < 0.01

Step 5: True or False: The evidence of a difference in the population mean scores is not significant at the 5% level.

True

False

Explanation / Answer

The anwers are -

Step 1: Explain why comparing children whose parents chose a Montessori school with children of other parents would not show whether Montessori schools perform better than other programs.

a. Because the subjects are not an SRS from all the children in the study.

Step 2: In all, 54 children were assigned to the Montessori school and 112 to other schools at age 3. When the children were 5, parents of 30 of the Montessori children and 25 of the others could be located and agreed to participate in testing. This information reveals a possible source of bias in the comparison of outcomes. Explain why.

a. Because participation in the testing was according to parents agreement.

Step 3. Difference in mean = 19 - 17 = 2

Standard error , SE = sqrt[ (s12/n1) + (s22/n2) ]

= sqrt[ (3.112/30) + (4.192/25) ] = 1.012249

t = difference in mean / SE = 2/1.012249 = 1.976

Step 4:

Degree of freedom, DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ] }

DF = (3.112/30 + 4.192/25)2 / { [ (3.112 / 30)2 / (30 - 1) ] + [ (4.192 / 25)2 / (25 - 1) ] }

= 43 (round to nearest integer)

P-value for t = 1.976, at DF = 43, is 0.0273

So, the answer is c. 0.025 < P < 0.05

Step 5: The evidence of a difference in the population mean scores is not significant at the 5% level. - False

It is significant as p-value < 0.05

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