Question

A researcher examined two factors that might predict self-esteem in college students: their grades in a Stats 2 class, and the number of followers they have on their Instagram account. She surveyed some 53 students and obtained the following values: SD_Y = 20, MS_M of 2000, and a significant F value. She would really love it if her model generalized outside of her original sample. What is her Adjusted R², and does the model still account for a significant amount of variance in other samples (Make sure to show your obtained F value, along with the degrees of freedom, to support your answer)?

Answer 1

Here we have N = 53, Standard deviation = 20, MSM = 2000 and k =2.

SSM = Sum of squares For Model

SSE = Sum of squares For Error

SST = Total Sum of squares

MSM = mean sum of squares for model

MSE = mean sum of squares for error

SDY²= total variation in the response y

Sources	SS	df	MS	F
Model	SSM	k-1	SSM/k-1	MSM/MSE
Error	SSE	N-k	SSE/N-k
Total	SST	N-1

Now k = 2 implies k-1 = 2-1 = 1.

Also, N-k = 53 - 2 = 51

Now, MSM = 2000 implies $MSM = \frac{SSM}{k-1} = \frac{SSM}{1}$

$2000 = \frac{SSM}{1}$

$SSM = 2000$

Now, SSM+ SSE = SST implies 2000 + SSE = SST

Also, Total variance = SD² = 400

Hence 2000 + SSE = 400 which implies SSE = 1600.

MSE =1600/51 = 31.37

F-value = MSM / MSE = 2000/47.06 = 63.75

ANOVA TABLE:

Sources	SS	df	MS	F
Model	2000	1	2000	63.75
Error	1600	51	31.37
Total	400	52	7.69

Adjusted R-squared value = $1-\frac{MSE}{MST} = 1-\frac{31.37}{7.69}=-3.08$