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Chapter 13: Problem 42

The California Department of Education assesses progress of \(\mathrm{K}-12\) students in meeting grade level standards in English/Language Arts and Mathematics yearly. A regression analysis was performed using 2016 assessment data from a sample of 195 California schools. The data consisted of the percentage of students at each school who met or exceeded the grade-level standards in English/Language Arts (SELA) and Mathematics (SMATH). Assume all conditions for the linear regression model hold. a. Using the information in the regression analysis output, is there a linear association between these two variables? Explain. b. Interpret the slope of the regression equation. c. Using a \(95 \%\) confidence interval, what is the estimated mean percent of mathematics proficient students at schools in which \(50 \%\) of students are proficient in English/Language Arts?

Short Answer

Expert verified
The specifics of the short answer can't be provided without the regression output, but general outlines of answers might be: a. Yes, there is strong/weak/no linear association between SELA and SMATH proficiency (based on \(R\)); b. For every one percent increase in proficiency in SELA, SMATH proficiency is expected to increase/decrease by the coefficient value; c. The estimated mean percentage of SMATH proficient students at schools where SELA proficiency is 50% lies within the calculated confidence interval.

Step by step solution

01

Checking linear association

To check if there's a linear association between the percentage of students who met the grade-level standards in English/Language Arts (SELA) and Mathematics (SMATH), we need to examine the correlation coefficient \(R\) in the regression output. A high absolute value of \(R\) (close to 1) indicates a strong linear association.
02

Interpreting the slope of the regression equation

The slope coefficient in the regression output represents the predicted change in the dependent variable (SMATH) for each one-unit change in the independent variable (SELA). Interpret this in the context of the data.
03

Calculating the estimated mean using confidence intervals

To calculate the estimated mean percent of mathematics proficient students at schools where 50% of students are proficient in English/Language Arts, use the regression equation (as provided or derived from the output), treat proficient English/Language Arts as 50%, and compute the corresponding proficient math percent. Then use this number and the standard error from the regression output to calculate the 95% CI.
04

Interpret results

Summarize the findings based on the three steps above and conclude the analysis. Interpret the association between proficiency in SELA and SMATH, what the slope represents and the estimated proficiency in Mathematics given the proficiency in English/Language Arts.

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