This exercise is based on summary statistics rather than raw data. This informat

Question

This exercise is based on summary statistics rather than raw data. This information is typically all that is presented in published reports. You can peform inference procedures by hand from the summaries. Use the conservative Option 2 (degrees of freedom the smaller of n1-1 and n2-1)Tor two-sample t confidence intervals and P-values. You must trust that the authors understood the conditions for inference and verified that they apply. This isn't always true. Do education programs for preschool children that follow the Montessori method perform better than other programs? A study compared five-year-old children in a certain city, who had been enrolled in preschool programs from the age of three. (a) 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. (In fact, all of the children in the study applied to the Montessori school. The school district assigned students to Montessori or other preschools by a random lottery.) O Parents who choose a Montessori school probably care more about their children than other parents O Parents who choose a Montessori school probably had an easier childhood than other parents. O Parents who choose a Montessori school probably have different attitudes about education than other parents. (b) In all, 54 children were assigned to the Montessori school and 112 to other schools at age three. When the children were five, parents of 39 of the Montessori children and 34 of the others could be located and agreed to and subsequently participated n testing. This in o mation reveals a possi e source o bias in the comparison o outcomes Explain y (Round your answers to the nearest whole number.) About % of Montessori parents participated in the study, compared to about % of the other parents. (c) 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: Group Montessori Control 30 19 3.23 25 17 4.12 Is there evidence of a difference in the population mean scores? (The researchers used two-sided alternative hypotheses.)

Explanation / Answer

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: 1 = 2
Alternative hypothesis: 1 2

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.10. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = sqrt[(s12/n1) + (s22/n2)]
SE = 1.01328
DF = 53
t = [ (x1 - x2) - d ] / SE

t = 1.974

where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 53 degrees of freedom is more extreme than -1.974; that is, less than -1.974 or greater than 1.974.

Thus, the P-value = 0.0536.

0.05 < P < 0.10

Interpret results. Since the P-value (0.0536) is greater than the significance level (0.05), we have to accept the null hypothesis.

There is not enough evidence to reject the hypothesis that Montessori school students perform differently than other school students in tests of problem solving using basic mathematics.

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