3.Consider the data set scores on a class of… | Linear Statistical Models End Term PYQ Dec 2023

Question 2

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3.Consider the data set scores on a class of 98 students in IIT-M. For each of 98 students, the composite score obtained in the class and the average number of hours studied per week is recorded. The following regression output was obtained using the scores data set

text

1Call:
2lm(formula = score ~ hours)
3
4Residuals:
5    Min      1Q  Median      3Q     Max
6-39.680 -14.675   0.215  14.088  54.785
7
8Coefficients:
9            Estimate Std. Error t value Pr(>|t|)
10(Intercept)   7.8805     4.6247   1.704   0.0916 .
11hours         7.1852     0.6134  11.713   <2e-16 ***
12---
13Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
14
15Residual standard error: 20.08 on 96 degrees of freedom
16Multiple R-squared:  0.5883,    Adjusted R-squared:  0.584
17F-statistic: 137.2 on 1 and 96 DF,  p-value: < 2.2e-16

For the following questions, please explain clearly which parts of the output are the basis for your answers. Include the units of variables wherever possible. Use the tabulated values in the cheat sheet.

(a) What is the predictor variable? What is the response variable? [2 Marks]

(b) Write the equation for the estimated conditional mean function, using the appropriate numerical values in the output. [2 Marks]

(c) Based on the estimated coefficients, can you give an estimate of $E[Y|X = 0]$ ? If yes, what is it (and show your work); if not, explain why not (missing information, inappropriate assumptions, etc.). [3 Marks]

(d) Give a 95% confidence interval for $\beta_1$ , assuming all the model assumptions hold. [2 Marks]

(e) What is $\hat{\sigma}^2$ , the in-sample mean-squared error? [3 Marks]

(f) From $n$ , the standard error of $\hat{\beta}_1$ and $\hat{\sigma}^2$ , can you find the sample variance in population across cities? If so, what is it? If not, explain. [5 Marks]

(g) Which part (or parts) of the output (if any) tests the assumption that the relationship between the predictor variable and the response variable is linear? [5 Marks]

A
I have written answers on the answer sheets
B
Not applicable

Question 2: 3.Consider the data set scores on a class of 98 students …

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