Q162SE Question: Improving Math SAT sco... [FREE SOLUTION]

Chapter 12: Q162SE (page 812)

Question: Improving Math SAT scores. Refer to the Chance (Winter 2001) study of students who paid a private tutor (or coach) to help them improve their Scholastic Assessment Test (SAT) scores, Exercise 2.88 (p. 113). Multiple regression was used to estimate the effect of coaching on SAT鈥揗athematics scores. Data on 3,492 students (573 of whom were coached) were used to fit the model $E (y) = 尾_{0} + 尾_{1} x_{1} + 尾_{2} x_{2}$ , where y = SAT-Math score, x₁ score on PSAT, and x₂{1 if the student was coached, 0 if not}.
The fitted model had an adjusted R² value of .76. Interpret this result.
The estimate of $尾_{2}$ in the model was 19, with a standard error of 3. Use this information to form a confidence interval for $尾_{2}$ . Interpret the interval.
Based on the interval, part b, what can you say about the effect of coaching on SAT鈥揗ath scores?
As an alternative model, the researcher added several 鈥渃ontrol鈥� variables, including dummy variables for student ethnicity (x₃,x₄ and x₅ ), a socioeconomic status index variable (x₆) , two variables that measured high school performance (x₇ and x₈₎ , the number of math courses taken in high school (x₉) , and the overall GPA for the math courses (x₁₀) . Write the hypothesized equation for E(y) for the alternative model.
Give the null hypothesis for a nested model F-test comparing the initial and alternative models.
The nested model F-test, part e, was statistically significant at . Practically interpret this result.
The alternative model, part d, resulted in $R_{a}^{2} = 0.79, {\hat{尾}}_{2} = 14 a n d s {\hat{尾}}_{2} = 3$ . Interpret the value of R²_a .
Refer to part g. Find and interpret a confidence interval for .
The researcher concluded that 鈥渢he estimated effect of SAT coaching decreases from the baseline model when control variables are added to the model.鈥� Do you agree? Justify your answer.
As a modification to the model of part d, the researcher added all possible interactions between the coaching variable (x₂) and the other independent variables in the model. Write the equation for E(y) for this modified model.
Give the null hypothesis for comparing the models, parts d and j. How would you perform this test?

Short Answer

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Answer

The value of R² in this question is 0.76, meaning that 76% of the variation in the data is explained by the model, indicating that the model is a good fit for the data.
confidence interval for $尾_{2}$ is (13.12, 24.88).
Since the confidence interval is a positive interval, it indicates that coached students have scored higher in SAT-Math.
The equation for E(y) for the alternate model can be written as $E (y) = 尾_{0} + 尾_{1} x_{1} + 尾_{2} x_{2} + 尾_{3} x_{3} + 尾_{4} x_{4} + 尾_{5} x_{5} + 尾_{6} x_{6} + 尾_{7} x_{7} + 尾_{8} x_{8} + 尾_{9} x_{9} + 尾_{10} x_{10}$ .
The null hypothesis can be written as $H_{0} : 尾_{3} = 尾_{4} = 尾_{5} = 尾_{6} = 尾_{7} = 尾_{8} = 尾_{9} = 尾_{10}$ while At least one of the parameters under test is non-zero.
It is given in the question that the nested model F-test is statistically significant at indicating $伪 = 0.05$ that the alternate model is a better fit for the data.
The alternate model has a R²_avalue of 0.79, indicating that 79% of the variation in the data is explained by the model making the model a good fit.
The confidence interval for is (9.12, 20.88).
The researcher鈥檚 conclusion that the estimated effect of SAT coaching decreases from the baseline model when a control variable is added to the model is incorrect. The alternate model was statistically significant, indicating that the added variables fit better.
The equation for E(y) for an alternate model with interaction can be written aslocalid="1662028198607" $E (y) = 尾_{0} + 尾_{1} x_{1} + 尾_{2} x_{2} + 尾_{3} x_{3} + 尾_{4} x_{4} + 尾_{5} x_{5} + 尾_{6} x_{6} + 尾_{7} x_{7} + 尾_{8} x_{8} + 尾_{9} x_{9} + 尾_{10} x_{10} + 尾_{11} x_{2} x_{1} + 尾_{12} x_{2} x_{3} + 尾_{13} x_{2} x_{4} + 尾_{14} x_{2} x_{5} + 尾_{15} x_{2} x_{6} + 尾_{16} x_{2} x_{7} + 尾_{17} x_{2} x_{8} + 尾_{18} x_{2} x_{9} + 尾_{19} x_{2} x_{10}$
To compare the two models, an F-test is conducted where the null and alternate hypothesis are $H_{0} : 尾_{11} = 尾_{12} = 尾_{13} = 尾_{14} = 尾_{15} = 尾_{16} = 尾_{17} = 尾_{18} = 尾_{19} = 0$ while H_a : At least one of the parameters under test is non-zero.

Step by step solution

(a) Interpretation of R2

The value of R² represents the fraction of the sample variance

of the y-values (measured by SS_yy ) that is explained by the least squares prediction equation.

R² And R²a have similar interpretations. However, unlike $R^{2}, R_{a}^{2}$ takes into account (鈥渁djusts鈥� for) both the sample size n and the number of b parameters in the model.

The value of R² in this question is 0.76, meaning that 76% of the variation in the data is explained by the model, indicating that the model is a good fit for the data.

(b) Confidence interval for β2

The confidence interval for $尾_{2}$ _颅can be written as $({\hat{尾}}_{2} 卤 t_{0.025, 3491} 脳 s {\hat{尾}}_{2})$

Therefore, the confidence interval for $尾_{2}$ is $(19 卤 1.96 脳 3)$

95% confidence interval for $尾_{2}$ is (13.12, 24.88).

(c) Effect of coaching on SAT-Math score

Since the confidence interval is positive, it indicates that coached students have scored higher in SAT-Math.

(d) Model equation for E(y)

The equation for E(y) for the alternate model can be written as

$E (y) = 尾_{0} + 尾_{1} x_{1} + 尾_{2} x_{2} + 尾_{3} x_{3} + 尾_{4} x_{4} + 尾_{5} x_{5} + 尾_{6} x_{6} + 尾_{7} x_{7} + 尾_{8} x_{8} + 尾_{9} x_{9} + 尾_{10} x_{10}$

(e) Null and alternate hypothesis

At least one of the parameters under test is non-zero.

$H_{0} : 尾_{3} = 尾_{4} = 尾_{5} = 尾_{6} = 尾_{7} = 尾_{8} = 尾_{9} = 尾_{10}$

(f) Evaluation of nested model F-test

It is given in the question that the nested model F-test is statistically significant at $伪 = 0.05$ indicating that the alternate model is a better fit for the data.

(g) Analysis of Ra2

The alternate model has a $R_{a}^{2}$ value of 0.79, indicating that 79% of the variation in the data is explained by the model making the model a good fit.

(h) Simplification for β2

The confidence interval for $尾_{2}$ _颅can be written as $({\hat{尾}}_{2} 卤 t_{0.025, 3491} 脳 s {\hat{尾}}_{2})$

Therefore, the confidence interval for $尾_{2}$ is

95% confidence interval for $尾_{2}$ is (9.12, 20.88).

(i) Significance of nested models

The researcher鈥檚 conclusion that the estimated effect of SAT coaching decreases from the baseline model when control variables are added to the model is not correct. The alternate model was statistically significant, indicating that the added variables fit better.

(j) Model equation with interaction terms

The equation for E(y) for the alternate model with interaction can be written as

$E (y) = 尾_{0} + 尾_{1} x_{1} + 尾_{2} x_{2} + 尾_{3} x_{3} + 尾_{4} x_{4} + 尾_{5} x_{5} + 尾_{6} x_{6} + 尾_{7} x_{7} + 尾_{8} x_{8} + 尾_{9} x_{9} + 尾_{10} x_{10} + 尾_{11} x_{2} x_{1} + 尾_{12} x_{2} x_{3} + 尾_{13} x_{2} x_{4} + 尾_{14} x_{2} x_{5} + 尾_{15} x_{2} x_{6} + 尾_{16} x_{2} x_{7} + 尾_{17} x_{2} x_{8} + 尾_{18} x_{2} x_{9} + 尾_{19} x_{2} x_{10}$

(k) Comparison of the nested model

To compare the two models, an F-test is conducted where the null and alternate hypotheses are $H_{0} : 尾_{11} = 尾_{12} = 尾_{13} = 尾_{14} = 尾_{15} = 尾_{16} = 尾_{17} = 尾_{18} = 尾_{19} = 0$ while

H_a: At least one of the parameters under test is non-zero.

Where, $F - test 鈥� statistic = \frac{S S E}{n - (k + 1)}$ .

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Short Answer

Step by step solution

(a) Interpretation of R2

(b) Confidence interval for β2

(c) Effect of coaching on SAT-Math score

(d) Model equation for E(y)

(e) Null and alternate hypothesis

(f) Evaluation of nested model F-test

(g) Analysis of Ra2

(h) Simplification for β2

(i) Significance of nested models

(j) Model equation with interaction terms

(k) Comparison of the nested model

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