Question

As concrete​ cures, it gains strength. The following data represent the​ 7-day and​ 28-day strength in pounds per square inch​ (psi) of a certain type of concrete. Complete parts​ (a) through​ (f) below.

​7-Day Strength​ (psi), x

24802480

33903390

23002300

33803380

26202620

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​28-Day Strength​ (psi), y

41204120

52205220

40704070

50205020

41904190

​(a) Treating the​ 7-day strength as the explanatory​ variable, x, use technology to determine the estimates of

beta 0β0

and

beta 1β1.

beta 0β0almost equals≈b 0b0equals=nothing

​(Round to one decimal place as​ needed.)

beta 1β1almost equals≈b 1b1equals=nothing

​(Round to four decimal places as​ needed.) ​(b) Compute the standard error of the​ estimate,

s Subscript ese.

s Subscript eseequals=nothing

​(Round to one decimal place as​ needed.)

​(c) A normal probability plot suggests that the residuals are normally distributed. Determine

s Subscript b 1sb1.

Use the answer from part

​(b).

s Subscript b 1sb1equals=nothing   

​(Round to four decimal places as​ needed.)

​(d) A normal probability plot suggests that the residuals are normally distributed. Test whether a linear relation exists between​ 7-day strength and​ 28-day strength at the

alphaαequals=0.050.05

level of significance.

State the null and alternative hypotheses. Choose the correct answer below.

A.

Upper H 0H0​:

beta 1β1equals=0

Upper H 1H1​:

beta 1β1not equals≠0

B.

Upper H 0H0​:

beta 0β0equals=0

Upper H 1H1​:

beta 0β0not equals≠0

C.

Upper H 0H0​:

beta 1β1equals=0

Upper H 1H1​:

beta 1β1greater than>0

D.

Upper H 0H0​:

beta 0β0equals=0

Upper H 1H1​:

beta 0β0greater than>0

Determine the​ P-value of this hypothesis test.

​P-valueequals=nothing

​(Round to three decimal places as​ needed.)

What is the conclusion that can be​ drawn?

A.

Do not rejectDo not reject

Upper H 0H0

and conclude that a linear relation

does not existdoes not exist

between the​ 7-day and​ 28-day strength of a certain type of concrete at the

alphaαequals=0.050.05

level of significance.

B.

Do not rejectDo not reject

Upper H 0H0

and conclude that a linear relation

existsexists

between the​ 7-day and​ 28-day strength of a certain type of concrete at the

alphaαequals=0.050.05

level of significance.

C.

RejectReject

Upper H 0H0

and conclude that a linear relation

existsexists

between the​ 7-day and​ 28-day strength of a certain type of concrete at the

alphaαequals=0.050.05

level of significance.

D.

RejectReject

Upper H 0H0

and conclude that a linear relation

does not existdoes not exist

between the​ 7-day and​ 28-day strength of a certain type of concrete at the

alphaαequals=0.050.05

level of significance.

​(e) Construct a​ 95% confidence interval about the slope of the true​ least-squares regression line.

Lower​ bound:

nothing

​(Round to three decimal places as​ needed.)

Upper​ bound:

nothing

​(Round to three decimal places as​ needed.)

​(f) What is the estimated mean​ 28-day strength of this concrete if the​ 7-day strength is 3000​ psi?

A good estimate of the mean​ 28-day strength is

nothing

psi

Answer 1

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.9821
R Square	0.9646
Adjusted R Square	0.9527
Standard Error	119.6316
Observations	5.0000
ANOVA
	df	SS	MS	F	Significance F
Regression	1.0000	11,68,384.8714	11,68,384.8714	81.6384	0.0029
Residual	3.0000	42,935.1286	14,311.7095
Total	4.0000	12,11,320.0000
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	1,553.5655	333.0802	4.6642	0.0186	493.5558	2,613.5752
7 day	1.0481	0.1160	9.0354	0.0029	0.6790	1.4173

a)

a)
b0=1553.5655
b1=1.0481
b)
se = 119.6316

c)
s_b1 = 0.1160

d)
Ho: b1 = 0
Ha: b1 not equal to 0

p-value = 0.0029 < 0.05
we reject the null hypothesis
there exists a relation between 7-day and 28-day