Question

A STAT 200 instructor believes that the average quiz score is a good predictor of final exam score. A random sample of 10 students produced the following data where x is the average quiz score and y is the final exam score.

x- 80, 95, 50, 60, 100, 55, 85, 70, 75, 85

y - 70, 96, 50, 63, 96, 60, 83, 60, 77, 87

(a) Find an equation of the least squares regression line. Round the slope and y-intercept value to two decimal places. Describe method for obtaining results. Show all work; writing the correct equation, without supporting work, will receive no credit. (b) Based on the equation from part (a), what is the predicted final exam score if the average quiz score is 65? Show all work and justify your answer. (c) Based on the equation from part (a), what is the predicted final exam score if the average quiz score is 40? Show all work and justify your answer. (d) Which predicted final exam score that you calculated for (b) and (c) do you think is closer to the true final exam score and why?

Answer 1

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
80	70	20.25	17.64	-18.90
95	96	380.25	475.24	425.10
50	50	650.25	585.64	617.10
60	63	240.25	125.44	173.60
100	96	600.25	475.24	534.10
55	60	420.25	201.64	291.10
85	83	90.25	77.44	83.60
70	60	30.25	201.64	78.10
75	77	0.25	7.84	-1.40
85	87	90.25	163.84	121.60

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	755	742	2522.5	2331.600	2304.000
mean	75.500	74.200	SSxx	SSyy	SSxy

a)

sample size ,   n =   10          
here, x̅ = Σx / n=   75.50   ,     ȳ = Σy/n =   74.20  
                  
SSxx =    Σ(x-x̅)² =    2522.5000          
SSxy=   Σ(x-x̅)(y-ȳ) =   2304.0          
                  
estimated slope , ß1 = SSxy/SSxx =   2304.0   /   2522.500   =   0.91338
                  
intercept,   ß0 = y̅-ß1* x̄ =   5.23984          
                  
so, regression line is   Ŷ =   5.24   +   0.91 *x

b)

Predicted Y at X=   65   is                  
Ŷ =   5.2398   +   0.9134   *   65   =   64.61

c)

Predicted Y at X=   40   is                  
Ŷ =   5.2398   +   0.9134   *   40   =   41.78

d)

predicted final exam score that calculated for (b) is closer to the true final exam score