Question

Length of exposure In weeks Hearing range In thousand cycles per second 47 15.1 56 14.1 116 13.2 178 12.7 19 14.6 75 13.8 160 11.9 31 14.8 12 15.3 164 12.6 43 14.7 74 14.0

e. Use your model to predict y when x is i)5, ii)100.

f. Provide 95% confidence intervals for the means (e, ii).

g. Provide 95% prediction intervals for (e, ii).

h. Comment on your findings in (f) and (g).

i. What were the basic assumptions for the model in (c)?

j. Would you consider the model in (b) to be a good model? Why or why not?

Answer 1

Length of exposure In weeks Hearing range In thousand cycles per second 47 15.1 56 14.1 116 13.2 178 12.7 19 14.6 75 13.8 160 11.9 31 14.8 12 15.3 164 12.6 43 14.7 74 14.0

Excel Addon Megastat used.

Menu used: correlation/Regression ---- Regression Analysis.

Regression Analysis
	r²	0.898	n	12
	r	-0.948	k	1
	Std. Error of Estimate	0.364	Dep. Var.	Hearing range
Regression output						confidence interval
variables		coefficients	std. error	t (df=10)	p-value	95% lower	95% upper
Intercept	a =	15.322
Length	b =	-0.017	0.002	-9.381	2.85E-06	-0.022	-0.013
ANOVA table
Source	SS	df	MS	F	p-value
Regression	11.691	1	11.691	88.00	2.85E-06
Residual	1.329	10	0.133
Total	13.020	11
Predicted values for: Hearingrange
		95% Confidence Intervals		95% Prediction Intervals
Length	Predicted	lower	upper	lower	upper	Leverage
5	15.234	14.840	15.629	14.332	16.137	0.236
100	13.572	13.325	13.819	12.723	14.421	0.093

The regression model is Hearing range(y) = 15.322-0.017*Length(x)

e. Use your model to predict y when x is i)5, ii)100.

predicted y when x is 5 = 15.234

predicted y when x is 100 = 13.572

f. Provide 95% confidence intervals for the means (e, ii).

95% CI for means= (13.325, 13.819)

g. Provide 95% prediction intervals for (e, ii).

95% PI for means= (12.723, 14.421)

h. Comment on your findings in (f) and (g).

95% PI for means is wider than 95% CI for means.

i. What were the basic assumptions for the model in (c)?

The four assumptions are:
Linearity of residuals
Independence of residuals
Normal distribution of residuals
Equal variance of residuals

j. Would you consider the model in (b) to be a good model? Why or why not?

The residual plots shows the patterns are not random and normal plot shows there is some violation of assumptions. The model may not be a good one.