Question

Values of modulus of elasticity (MOE, the ratio of stress, i.e., force per unit area, to strain, i.e., deformation per unit length, in GPa) and flexural strength (a measure of the ability to resist failure in bending, in MPa) were determined for a sample of concrete beams of a certain type, resulting in the following data: MOE 29.9 33.4 33.6 35.3 35.4 36.2 36.3 36.5 37.7 37.9 38.6 38.8 39.7 41.2 Strength 6.0 7.1 7.3 6.1 8.0 6.6 6.8 7.7 6.7 6.7 7.0 6.5 8.1 8.8 MOE 42.8 42.8 43.4 45.8 45.8 47.0 48.1 49.2 51.8 62.6 69.9 79.6 80.2 Strength 8.3 8.8 8.0 9.6 7.6 7.5 9.7 7.7 7.5 11.6 11.5 11.8 10.9 Fitting the simple linear regression model to the n = 27 observations on x = modulus of elasticity and y = flexural strength given in the data above resulted in ŷ = 7.576, sy hat = 0.178 when x = 40 and ŷ = 9.777, sy hat = 0.251 for x = 60. (a) Explain why sy hat is larger when x = 60 than when x = 40. The closer x is to x, the smaller the value of sy hat. The farther x is from y, the smaller the value of sy hat. The farther x is from x, the smaller the value of sy hat. The closer x is to y, the smaller the value of sy hat. (b) Calculate a confidence interval with a confidence level of 95% for the true average strength of all beams whose modulus of elasticity is 40. (Round your answers to three decimal places.) , MPa (c) Calculate a prediction interval with a prediction level of 95% for the strength of a single beam whose modulus of elasticity is 40. (Round your answers to three decimal places.) , MPa (d) If a 95% CI is calculated for true average strength when modulus of elasticity is 60, what will be the simultaneous confidence level for both this interval and the interval calculated in part (b)? The simultaneous confidence level for these intervals is at least

Answer 1

Values of modulus of elasticity (MOE, the ratio of stress, i.e., force per unit area, to strain, i.e., deformation per unit length, in GPa) and flexural strength (a measure of the ability to resist failure in bending, in MPa) were determined for a sample of concrete beams of a certain type, resulting in the following data: MOE 29.9 33.4 33.6 35.3 35.4 36.2 36.3 36.5 37.7 37.9 38.6 38.8 39.7 41.2 Strength 6.0 7.1 7.3 6.1 8.0 6.6 6.8 7.7 6.7 6.7 7.0 6.5 8.1 8.8 MOE 42.8 42.8 43.4 45.8 45.8 47.0 48.1 49.2 51.8 62.6 69.9 79.6 80.2 Strength 8.3 8.8 8.0 9.6 7.6 7.5 9.7 7.7 7.5 11.6 11.5 11.8 10.9 Fitting the simple linear regression model to the n = 27 observations on x = modulus of elasticity and y = flexural strength given in the data above resulted in ŷ = 7.576, sy hat = 0.178 when x = 40 and ŷ = 9.777, sy hat = 0.251 for x = 60.

1. Explain why sy hat is larger when x = 60 than when x = 40.

The closer x is to x, the smaller the value of sy hat.

(b) Calculate a confidence interval with a confidence level of 95% for the true average strength of all beams whose modulus of elasticity is 40. (Round your answers to three decimal places.) , MPa

95% CI = (7.209, 7.943)

Calculate a prediction interval with a prediction level of 95% for the strength of a single beam whose modulus of elasticity is 40. (Round your answers to three decimal places.) , MPa

95% PI = (5.764, 9.387)

(d) If a 95% CI is calculated for true average strength when modulus of elasticity is 60, what will be the simultaneous confidence level for both this interval and the interval calculated in part (b)? The simultaneous confidence level for these intervals is at least

95% CI =(9.260, 10.294) and the is wider.

Regression Analysis
	r²	0.750	n	27
	r	0.866	k	1
	Std. Error	0.861	Dep. Var.	Strength
ANOVA table
Source	SS	df	MS	F	p-value
Regression	55.5608	1	55.5608	74.90	5.44E-09
Residual	18.5459	25	0.7418
Total	74.1067	26
Regression output					confidence interval
variables	coefficients	std. error	t (df=25)	p-value	95% lower	95% upper
Intercept	3.1730	0.5979	5.307	1.69E-05	1.9416	4.4044
MOE	0.1101	0.0127	8.654	5.44E-09	0.0839	0.1363
Predicted values for: Strength
		95% Confidence Intervals		95% Prediction Intervals
MOE	Predicted	lower	upper	lower	upper	Leverage
40	7.5758	7.2085	7.9430	5.7643	9.3872	0.043
60	9.7771	9.2599	10.2944	7.9294	11.6249	0.085