Math / Data & Statistics

QuestionCONFIDENTIAL DEC2024/ECO545
PART B (30 MARKS) QUESTION 1 Farah Ahmad is studying the relationship between students' final exam scores $Y_{i}$ (measured in GPA) and their hours spent studying per week $X_{i}$ (measured in hours). The population regression equation is given by: $Y i=\beta_{0}+\beta_{1} X_{i}+\mu$
Preliminary analysis of the sample data produces the following statistics: $\begin{array}{l} \sum Y=48095 \quad \sum X=95050 \quad \sum Y^{2}=406538 \quad \sum e=5807 \\ \quad \sum X^{2}=354446 \quad \sum X Y=1554698 \quad n=1200 \end{array}$
Use the above information to answer the following questions. a) Estimate the regression slope and intercept (6 marks) b) Write the estimated regression equation (1 mark) c) Interpret the estimated slope coefficient. (2 marks) d) Compute the coefficient determination and interpret it. (4 marks) e) Predict the exam score if the student spends 5 hours per week of studying. (2 marks)

Studdy Solution

STEP 1

1. The population regression equation is $Y_i = \beta_0 + \beta_1 X_i + \mu$ .
2. The sample size $n = 1200$ .
3. The provided sums are for the sample data.

STEP 2

1. Estimate the regression slope ( $\beta_1$ ) and intercept ( $\beta_0$ ).
2. Write the estimated regression equation.
3. Interpret the estimated slope coefficient.
4. Compute the coefficient of determination ( $R^2$ ) and interpret it.
5. Predict the exam score for 5 hours of study per week.

STEP 3

Calculate the regression slope ( $\beta_1$ ) using the formula:
$\beta_1 = \frac{\sum (X_i Y_i) - \frac{\sum X_i \sum Y_i}{n}}{\sum (X_i^2) - \frac{(\sum X_i)^2}{n}}$
Substitute the given values:
$\beta_1 = \frac{1554698 - \frac{95050 \times 48095}{1200}}{354446 - \frac{95050^2}{1200}}$
Calculate the result:
$\beta_1 = \frac{1554698 - 3807937.5}{354446 - 7527087.5}$
$\beta_1 = \frac{-2253239.5}{-7172641.5} \approx 0.314$

STEP 4

Calculate the regression intercept ( $\beta_0$ ) using the formula:
$\beta_0 = \frac{\sum Y_i}{n} - \beta_1 \frac{\sum X_i}{n}$
Substitute the given values:
$\beta_0 = \frac{48095}{1200} - 0.314 \times \frac{95050}{1200}$
Calculate the result:
$\beta_0 = 40.0792 - 0.314 \times 79.2083$
$\beta_0 = 40.0792 - 24.857 \approx 15.222$

STEP 5

Write the estimated regression equation:
$\hat{Y_i} = 15.222 + 0.314 X_i$

STEP 6

Interpret the estimated slope coefficient:
The slope coefficient ( $\beta_1 = 0.314$ ) indicates that for each additional hour spent studying per week, the student's final exam score is expected to increase by approximately 0.314 GPA points, holding all else constant.

STEP 7

Compute the coefficient of determination ( $R^2$ ) using the formula:
$R^2 = \frac{\left(\sum (X_i Y_i) - \frac{\sum X_i \sum Y_i}{n}\right)^2}{\left(\sum (X_i^2) - \frac{(\sum X_i)^2}{n}\right) \left(\sum (Y_i^2) - \frac{(\sum Y_i)^2}{n}\right)}$
Substitute the given values and calculate:
$R^2 = \frac{(-2253239.5)^2}{(-7172641.5)(406538 - \frac{48095^2}{1200})}$
Calculate the result:
$R^2 \approx 0.098$
Interpretation: Approximately 9.8% of the variability in the final exam scores can be explained by the hours spent studying per week.

STEP 8

Predict the exam score for a student studying 5 hours per week:
$\hat{Y} = 15.222 + 0.314 \times 5$
Calculate the result:
$\hat{Y} = 15.222 + 1.57 = 16.792$
The predicted exam score for a student studying 5 hours per week is approximately 16.792 GPA points.

Was this helpful?

Studdy Solution

STEP 1

1. The population regression equation is $Y_i = \beta_0 + \beta_1 X_i + \mu$ .
2. The sample size $n = 1200$ .
3. The provided sums are for the sample data.

STEP 2

1. Estimate the regression slope ( $\beta_1$ ) and intercept ( $\beta_0$ ).
2. Write the estimated regression equation.
3. Interpret the estimated slope coefficient.
4. Compute the coefficient of determination ( $R^2$ ) and interpret it.
5. Predict the exam score for 5 hours of study per week.

STEP 3

STEP 4

STEP 5

Write the estimated regression equation:
$\hat{Y_i} = 15.222 + 0.314 X_i$

STEP 6

Interpret the estimated slope coefficient:
The slope coefficient ( $\beta_1 = 0.314$ ) indicates that for each additional hour spent studying per week, the student's final exam score is expected to increase by approximately 0.314 GPA points, holding all else constant.

STEP 7

STEP 8

Predict the exam score for a student studying 5 hours per week:
$\hat{Y} = 15.222 + 0.314 \times 5$
Calculate the result:
$\hat{Y} = 15.222 + 1.57 = 16.792$
The predicted exam score for a student studying 5 hours per week is approximately 16.792 GPA points.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

1. The population regression equation is Yi=β0+β1Xi+μ Y_i = \beta_0 + \beta_1 X_i + \mu Yi​=β0​+β1​Xi​+μ.2. The sample size n=1200 n = 1200 n=1200.3. The provided sums are for the sample data.

STEP 2

STEP 3

STEP 4

STEP 5

Write the estimated regression equation:Yi^=15.222+0.314Xi\hat{Y_i} = 15.222 + 0.314 X_iYi​^​=15.222+0.314Xi​

STEP 6

Interpret the estimated slope coefficient:The slope coefficient (β1=0.314\beta_1 = 0.314β1​=0.314) indicates that for each additional hour spent studying per week, the student's final exam score is expected to increase by approximately 0.314 GPA points, holding all else constant.

STEP 7

STEP 8

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The population regression equation is $Y_i = \beta_0 + \beta_1 X_i + \mu$ .
2. The sample size $n = 1200$ .
3. The provided sums are for the sample data.

Write the estimated regression equation:
$\hat{Y_i} = 15.222 + 0.314 X_i$

Interpret the estimated slope coefficient:
The slope coefficient ( $\beta_1 = 0.314$ ) indicates that for each additional hour spent studying per week, the student's final exam score is expected to increase by approximately 0.314 GPA points, holding all else constant.