Question

A mathematics achievement test is given to students prior to entering a certain college. A sample of 10 students was selected and their progress in calculus observed:

Student	Achievement Test Score, X	Final Calculus Grade, Y
1	39	65
2	43	78
3	21	52
4	64	82
5	57	92
6	47	89
7	28	73
8	75	98
9	34	56
10	52	75

e) Complete the ANOVA table for the least squares estimate for the regression line.

f)  Test at the 5% significance level whether the model is useful for predicting the final calculus grade.

g)  Give a 95% confidence interval for the mean final calculus grade when the achievement test score is 50.

h) Give a 95% prediction interval for the final calculus grade when the achievement test score is 50. Is the interval wider or narrower than the confidence interval? Why?

Answer 1

E.

Excel output for the Anova Table :

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.839785887
R Square	0.705240336
Adjusted R Square	0.668395378
Standard Error	8.703633358
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	1	1449.974131	1449.974131	19.1407556	0.002364532
Residual	8	606.025869	75.75323363
Total	9	2056
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	40.78415521	8.506861379	4.794265875	0.00136551	21.1672977	60.40101273	21.1672977	60.40101273
Achievement Test Score, X	0.765561843	0.174984967	4.375014926	0.002364532	0.362045785	1.169077901	0.362045785	1.169077901

F.

H₀: β₁ = 0 versus H_A: β₁ ≠ 0:

As we can see p-value for the model is 0.00236 < 0.05 (At 5% confidence) i. e. H₀ will be rejected and hence we can say that the model is significant and can be used for predicting the final calculus grade.

G.

Regression Line:

y = 40.78 + 0.766x

y = 40.78 + 0.766*50

= 79.0622

Margin of Error: (By inputting values in above formula)

E = 6.548

So, the Confidence Interval is:

79.0622 6.548

= [72.514 ,85.61]

H.

Prediction Confidence interval :

As compared with the expected value interval, the prediction interval is wider because :

The standard error for a confidence interval on the mean takes into account the uncertainty due to sampling. The line you computed from your sample will be different from the line that would have been computed if you had the entire population, the standard error takes this uncertainty into account.

The standard error for a prediction interval on an individual observation takes into account the uncertainty due to sampling like above but also takes into account the variability of the individuals around the predicted mean. The standard error for the prediction interval will be wider than for the confidence interval and hence the prediction interval will be wider than the confidence interval.

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