Overview

This assignment asks you to examine, for children from households below the poverty level, whether there is an achievement gap in child math scale score at kindergarten entry (C1MSCALE) associated with child language minority status (LANGMINO) or primary home language (HOMELANG) and whether mother’s education in years (MOMEDYEAR) explains the achievement gap. By restricting our interest to the population of poor children, we intend to isolate the effect of mother’s education from the effect of household income (please note that I did not include the income measure in the data file because it becomes irrelevant to the current study).

First of all, you are asked to examine whether mother’s education predicts child math preparation at kindergarten entry. You will assess the math achievement gap between language minority children and English-speaking children at kindergarten entry. You will then consider whether language minority children are disadvantaged in math preparation due to a lower level of mother’s education on average.

Subsequently, after separating the children of Spanish-speaking families from other language minority children, you will re-examine the achievement gap between Spanish-speaking children and English-speaking children and that between other non-English-speaking children and English-speaking children. You will further determine whether the achievement gaps in math preparation remain among different language groups that cannot be attributed to between-group differences in mother’s average education level.

In the end, you will be asked to evaluate the statement that raising maternal education level could potentially overcome the disadvantage in math preparation experienced by Spanish- speaking children.

The assignment has four parts including 21 questions in total.

Part I. Does Mother’s Education Predict Student Math Skills at Kindergarten Entry?

1. Graphically display the relationship between mother’s education in years and child math scale score at kindergarten entry. What kind of relationship do you see? (Consider the following aspects of a relationship: linear vs. non-linear, positive vs. negative). For what kind of data is linear regression a suitable technique? Does it make sense to use simple linear regression in this case? Why or why not?
2. Write down the population model for using mother’s education in years to predict child math scale score at kindergarten entry. Define each term in the model. Identify the population parameters to be estimated. Use SPSS to obtain estimates of the parameters. Provide an interpretation for each of the estimates.
3. Pose the null and alternative hypotheses regarding the slope in the model. Under the null hypothesis, the slope estimator will have a definite sampling distribution. What do you know about that distribution with regard to its central tendency, dispersion, and shape? How do you reach a decision about the null hypothesis? What is the meaning of the p-value? Compute a 95% confidence interval for the slope. Explain what this interval tells us.
4. What is your conclusion about the relationship between mother’s education and child math skills at kindergarten entry?
   Part II. What Is the Math Achievement Gap between Language Minority Children and English- Speaking Children at Kindergarten Entry?
5. Use language minority status as a predictor of child math scale score at kindergarten entry in a simple linear regression model. Write down the population model for the regression. Define each term in the model. Obtain estimates of the parameters. Provide an interpretation for each of the estimates.
6. What does the t statistic for the slope indicate? What does the F statistic indicate?
7. What is your conclusion about the relationship between language minority status and math preparation at kindergarten entry?
   Part III. Are Language Minority Children Disadvantaged in Math Preparation due to Lower Maternal Education?
8. Report the sample mean difference in mother’s education in years between language minority students and English-speaking students. Use a two independent sample t test to draw inference about the population mean difference between the two groups.
9. Include both language minority status and mother’s education in years as predictors of child math scale score in a multiple regression model. Write down the population model and define each term in the model. Pose the null and the alternative hypotheses regarding the model as a whole. Then pose the null and alternative hypotheses associated with each regression coefficient.
10. Obtain estimates of the parameters. Provide an interpretation for each of the estimates. Make 2 decisions about the null hypotheses.
11. Compare with the results for Question 5. Report the changes in the intercept estimate, in the slope estimate for language minority status, and in the estimated variance of residual. Explain these differences.
12. Do you think the math achievement gap at kindergarten entry between language minority children and English-speaking children is at least partly attributable to the difference in mother’s average education level between these two groups? Why or why not?
13. Save the predicted math scale score on the basis of the multiple regression model. Compute the correlation between the observed math scale score and the predicted math scale score. Square the correlation and define its meaning.
14. Save the residual from the multiple regression model. Compare the variance of the residual and the variance of the observed math scale score by computing their ratio. What does the ratio indicate to us about the adequacy of the model?
15. What statistical assumptions are needed to justify the use of the t or F test in regression? Graph the univariate distribution of the residual in a histogram. Also graph the bivariate distribution of the predicted value and the residual in a scatter plot. Examine these distributions and explain the implications for evaluating the tenability of the assumptions.
    Part IV. A Close Look at the Math Achievement Gap Associated with Primary Home Language
16. Compare average child math scale score at kindergarten entry among English-speaking children, Spanish-speaking children, and children from other non-English-speaking households in the sample. Conduct one-way Analysis of Variance followed by post-hoc comparisons to determine whether the achievement gaps exist among these three language groups in the population.
17. Compare mother’s average education level among the three language groups in the sample. Conduct one-way Analysis of Variance followed by post-hoc comparisons to determine whether the three groups differ in mother’s average education level in the population.
18. Create two dummy indicators, one for Spanish-speaking children, and the second for other non-English-speaking children. Analyze a multiple regression model using these two dummy indicators along with mother’s education in years to predict child math scale score at kindergarten entry. Write down the population model and define each term in the model. Obtain estimates of the parameters. Provide an interpretation for each of the estimates.
19. Is there a math achievement gap at kindergarten entry between Spanish-speaking children and 3 English-speaking children? Comparing Spanish-speaking children and English-speaking children whose mothers had the same years of education, is the achievement gap reduced between these two groups? What are possible explanations for the remaining gap?
20. Is there a math achievement gap at kindergarten entry between other non-English-speaking children and English-speaking children? Comparing other non-English-speaking children and English-speaking children whose mothers had the same years of education, is the achievement gap reduced between these two groups? What are possible explanations for the remaining gap?
21. Based on the above evidence, would you argue that raising mother’s education level could potentially overcome the disadvantage in math preparation experienced by Spanish-speaking children? Why or why not?

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