Question

I need help with - What is the conclusion that can be​ drawn? - thank you!

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.

7-Day_Strength_(psi)_-_x   28-Day_Strength_(psi)_-_y
3390   5220
3340   4630
2300   4070
2480   4120
3380   5020

​(a) Treating the​ 7-day strength as the explanatory​ variable, x, use technology to determine the estimates of β0 and β1.

β0≈b0= 1981.5

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

β1≈b1= 0.8833

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

​(b) Compute the standard error of the​ estimate se.

se=230.5

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

(c) A normal probability plot suggests that the residuals are normally distributed. Determine sb1. Use the answer from part ​(b).

sb1=0.2131  

​(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 α=0.05 level of significance. State the null and alternative hypotheses. Choose the correct answer below.

A.H0​: β1=0 H1​: β1>0

B. H0​: β0=0 H1​: β0≠0

C. H0​: β1=0 H1​: β1≠0 correct answer

D. H0​: β0=0 H1​: β0>0

Determine the​ P-value of this hypothesis test.

​P-value =0.026

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

What is the conclusion that can be​ drawn?

A.

Reject H0 and conclude that a linear relation exists between the​ 7-day and​ 28-day strength of a certain type of concrete at the α=0.05 level of significance.

B.

Do not reject H0 and conclude that a linear relation exists between the​ 7-day and​ 28-day strength of a certain type of concrete at the α=0.05level of significance.

C.

Reject H0 and conclude that a linear relation does not exist between the​ 7-day and​ 28-day strength of a certain type of concrete at the α=0.05 level of significance.

D.

Do not reject H0 and conclude that a linear relation does not exist between the​ 7-day and​ 28-day strength of a certain type of concrete at the α=0.05 level of significance.

Answer 1

Here null hypothesis H_{​​​​​​0} : $\beta$1 = 0

( linear relation does not exist between 7 day and 28 day strength of certain type of concrete)

Alternative hypothesis H_​1 : $\beta$1 $\neq$ 0

( linear relation exists between 7 day and 28 day strength of certain type of concrete )

Test criteria : If p-value < $\alpha$ we reject the null hypothesis H0 , otherwise we fail to reject the null hypothesis .

Here $\alpha$ = 0.05

Our p value = 0.026 <  0.05   

Conclusion : Reject H0 and conclude that a linear relation exists between the 7 day and 28 days strength of certain type of concrete at $\alpha$ = 0.05 level if significance.