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The Perry Preschool Project was created in the early \(1960 \mathrm{~s}\) by David Weikart in Ypsilanti, Michigan. In this project, 123 African American children were randomly assigned to one of two groups: One group enrolled in the Perry Preschool, and the other group did not. Follow-up studies were done for decades. One research question was whether attendance at preschool had an effect on high school graduation. The table shows whether the students graduated from regular high school or not and includes both boys and girls (Schweinhart et al. 2005 ). Find a \(95 \%\) confidence interval for the difference in proportions, and interpret it. $$ \begin{array}{|lcc|} \hline & \text { Preschool } & \text { No Preschool } \\ \hline \text { Grad HS } & 37 & 29 \\ \hline \text { No Grad HS } & 20 & 35 \\ \hline \end{array} $$

Step by step solution

01

Identifying data

First, identify the data given. For the 'Preschool' group: 37 graduated from high school (call this \(x_1\)), and altogether there were 57 students (37+20, call this \(n_1\)). For the 'No Preschool' group: 29 graduated from high school (call this \(x_2\)), and altogether there were 64 students (29+35, call this \(n_2\)).

02

Calculate proportions

Next, calculate the proportions of graduations in both groups using the formula \(\hat{p}=x/n\). The proportion of graduations in 'Preschool' group is \( \hat{p}_1 = x_1/n_1 = 37/57 = 0.649 \). The proportion of graduations in 'No Preschool' group is \( \hat{p}_2 = x_2/n_2 = 29/64 = 0.453 \). The difference \(d\) between these two proportions is \(d = \hat{p}_1 - \hat{p}_2 = 0.649 - 0.453 = 0.196 \).

03

Calculate the standard error

Next, calculate the standard error (SE) using the formula \(SE = \sqrt{ \hat{p}_1(1 - \hat{p}_1)/n_1 + \hat{p}_2(1 - \hat{p}_2)/n_2 }\). Here, the SE would be \(SE = \sqrt{ 0.649(1 - 0.649)/57 + 0.453(1 - 0.453)/64 } = 0.093\).

04

Constructing 95% confidence interval

Construction of a 95% confidence interval involves the formula \(d \pm Z_{\text{crit}} \times SE\). Here \(Z_{\text{crit}}\) is the critical value based on the normal distribution (use 1.96 for a 95% confidence level). The confidence interval therefore is \(0.196 \pm 1.96 \times 0.093\), which approximate to \(0.196 \pm 0.182\). Hence, the confidence interval is (0.014, 0.378).

05

Interpret the confidence interval

The interpretation would be: We are 95% confident that the difference in proportions of high school graduation between the Perry Preschool group and the 'No Preschool' group is between 1.4% and 37.8%. That is, the Perry Preschool increases the graduation rate by anything from 1.4% to 37.8% compared to 'No Preschool'.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perry Preschool Project

The Perry Preschool Project was a significant educational study initiated in the 1960s, focusing on the impact of early childhood education on African American children in Ypsilanti, Michigan. The study involved a comparison between children who attended the Perry Preschool and those who did not. Over several decades, researchers conducted follow-up studies to assess long-term effects on various outcomes, including high school graduation rates. By examining different aspects of the participants' lives, the study aimed to provide evidence on the value of preschool education.

Proportion Calculation

Proportion calculation is a fundamental concept in statistics used to measure the size of a subset relative to the whole population. For the Perry Preschool Project, this involved determining the proportion of students who graduated from high school in both the preschool and no-preschool groups. The formula for calculating a proportion is simple: \( \hat{p} = \frac{x}{n} \) where \( \hat{p} \) is the estimated proportion, \( x \) is the number of successes (in this case, graduations), and \( n \) is the total number of observations. By comparing these proportions, researchers could better understand the impact of preschool education on high school graduation rates.

Standard Error Calculation

Standard error (SE) is a critical statistical concept that measures the variability of an estimate. In proportion calculations like those in the Perry Preschool Project, SE helps in understanding the precision of the proportion estimates. It is calculated using the formula: \( SE = \sqrt{\hat{p}_1(1 - \hat{p}_1)/n_1 + \hat{p}_2(1 - \hat{p}_2)/n_2} \) where \( \hat{p}_1 \) and \( \hat{p}_2 \) are the estimated proportions, and \( n_1 \) and \( n_2 \) are the sizes of the two groups. A lower SE indicates a more reliable estimate of the proportion. It is of paramount importance in computing confidence intervals, which provide a range of values within which the true population parameter is expected to lie.

Statistical Significance

Statistical significance is a determination of whether the observed effects in data are likely due to the relationship between variables rather than random chance. In the context of the Perry Preschool Project, determining statistical significance would help reveal if the difference in high school graduation rates between the preschool and no-preschool groups can be attributed to the preschool intervention rather than being a random variation. Typically, a test statistic is compared against a threshold value (like 1.96 for a 95% confidence level). If the test statistic exceeds this threshold, the results are deemed statistically significant, indicating that the effects observed (e.g., higher graduation rates) are probably not occurring by chance alone.