Chapter 10: Q.3.1 (page 619)

To study the long-term effects of preschool programs for poor children, researchers designed an experiment. They recruited $123$ children who had never attended preschool from low-income families in Michigan. Researchers randomly assigned $62$ of the children to attend preschool (paid for by the study budget) and the other $61$ to serve as a control group who would not go to preschool. One response variable of interest was the need for social services for adults. In the past $10$ years, $38$ children in the preschool group and $49$ in the control group have needed social services.⁴ $1$ . Does this study provide convincing evidence that preschool reduces the later need for social services? Carry out an appropriate test to help answer this question.

Short Answer

Expert verified

From the given information, the probability is $0.2061$

Step by step solution

Given Information

It is given in the question that, from the previous exercise

$Mean (μ_{{\hat{p}}_{1} - {\hat{p}}_{2}}) = - 0.10$

$Standard deviation (σ_{{\hat{p}}_{1} - {\hat{p}}_{2}}) = 0.097$

Does this study provide convincing evidence that preschool reduces the later need for social services?

Explanation

$The probability that {\hat{p}}_{1} - {\hat{p}}_{2} is less than or equal to - 0.02 can be computed as:$

localid="1650435144025" $P ({\hat{p}}_{1} - {\hat{p}}_{2} \geq - 0.02) = P (\frac{({\hat{p}}_{1} - {\hat{p}}_{2}) - (p_{1} - p_{2})}{(σ_{{\hat{p}}_{1} - {\hat{p}}_{2}})} \geq \frac{- 0.02 - (p_{1} - p_{2})}{(σ_{{\hat{p}}_{1} - {\hat{p}}_{2}})}) = P (Z \geq \frac{- 0.02 - (- 0.1)}{0.097}) = P (Z > 0.82) = 0.2061$