Question

7-Day Strength (psi) x: 3380 2620 2890 3390 2480

28-Day Strength (psi) y: 5020 4190 4620 5220 4120

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.

(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=? (Round to one decimal 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 e.

s Subscript e equals=? (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=?  ​(Round to four decimal places as​ needed.)

Determine P-value of this hypothesis test.

P-value=? (Round to three decimal places as needed)

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

Lower​ bound: ? ​(Round to three decimal places as​ needed.)

Upper​ bound: ? ​(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 ? psi. ​(Round to two decimal places as​ needed.)

Answer 1

a)

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	14760.00	18170.00	711880.00	15093920.00	-1377240.00
mean	2952.00	3634.00	SSxx	SSyy	SSxy

sample size ,   n =   5          
here, x̅ = Σx / n=   2952.000   ,     ȳ = Σy/n =   3634.000  
                  
SSxx =    Σ(x-x̅)² =    711880.0000          
SSxy=   Σ(x-x̅)(y-ȳ) =   -1377240.0          
                  
estimated slope , ß1 = SSxy/SSxx =   -1377240.0   /   711880.000   =   -1.9347
                  
intercept,   ß0 = y̅-ß1* x̄ =   9345.09243          
                  
so, regression line is   Ŷ =   9345.092   +   -1.935   *x
.................

b)

SSE=   (SSxx * SSyy - SS²xy)/SSxx =    12429440.007
      
std error ,Se =    √(SSE/(n-2)) =    2035.4721

..................

c)

slope hypothesis test               tail=   2
Ho:   ß1=   0          
H1:   ß1╪   0          
n=   5              
alpha =   0.05              
estimated std error of slope =Se(ß1) = Se/√Sxx =    2035.472   /√   711880.00   =   2.4125
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    -1.9347   /   2.4125   =   -0.8019
                     
Degree of freedom ,df = n-2=   3              
p-value =    0.481

..............

e)

confidence interval for slope                  
α=   0.05              
t critical value=   t α/2 =    3.182   [excel function: =t.inv.2t(α/2,df) ]      
estimated std error of slope = Se/√Sxx =    2035.47210   /√   711880.00   =   2.412
                  
margin of error ,E= t*std error =    3.182   *   2.412   =   7.678
estimated slope , ß^ =    -1.9347              
                  
                  
lower confidence limit = estimated slope - margin of error =   -1.9347   -   7.678   =   -9.612
upper confidence limit=estimated slope + margin of error =   -1.9347   +   7.678   =   5.743
.......

f)

Predicted Y at X=   3000   is          
Ŷ =   9345.0924   +   -1.9347   *3000=   3541.14

..................

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