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) for 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 had an easier childhood than other parents. Parents who choose a Montessori school probably care more about their children than other parents. Parents who choose a Montessori school probably have different attitudes about education than other parents. (b) In all, 55 children were assigned to the Montessori school and 116 to other schools at age three. When the children were five, parents of 37 of the Montessori children and 31 of the others could be located and agreed to and subsequently participated in testing. This information reveals a possible source of bias in the comparison of outcomes. Explain why. (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 х Montessori 30 19 3.12 Control 25 17 4.22 Is there evidence of a difference in the population mean scores? (The researchers used two-sided alternative hypotheses.) PLAN: Let u, be the mean score for children in Montessori schools, and let u, be the mean score for children in the control schools. State the null and alternative hypotheses. О Но: И1 3D А2 Ha: H1 + H2 О Но: Н1 > М2 Ha: H1 # µ2 О Но: 11 + И2 Ha: H1 = H2 о Но: Н1 %3D А2 На: Н1 > Н2 SOLVE: Calculate the test statistic. (Round your answer to three decimal places.) t =