Question

The accompanying data table contains the listed prices and weights of the diamonds in

30

rings offered for sale in a newspaper. The prices are in​ dollars, with the weights in carats. Formulate the regression model with price as the response and weight as the explanatory variable. Complete parts​ (a) and​ (b) below.

​(a) Could these data be a sample from a population in which the population intercept is​ zero? Should

β0=​0?

Conduct a hypothesis test for

β0.

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

A.

H0: β0=0​,

Ha: β0<0

B.

H0: β0=0​,

Ha: β0≠0

C.

H0: β0≠0​,

Ha: β0=0

D.

H0: β0≥0​,

Ha: β0<0Find the value of the​ t-statistic for

β0.

What is the value of the test​ statistic?

t=

​(Type an integer or decimal rounded to three decimal places as​ needed.)

Identify the​ p-value of this test.

​p-value=

​(Type an integer or decimal rounded to three decimal places as​ needed.)

Compare the​ p-value to 0.05. Choose the correct conclusion below.

A.Reject

H0.

There is sufficient evidence that the population intercept is not zero.

B.Fail to reject

H0.

There is insufficient evidence that the population intercept is not zero.

C.Fail to reject

H0.

There is sufficient evidence that the population intercept is not zero.

D.Reject

H0.

There is insufficient evidence that the population intercept is not zero. Should

β0=​0?

A.

​No, because one would expect a negative​ intercept, representing the variable cost of the ring.

B.

​Yes, because one would expect an intercept representing the fixed cost of the ring.

C.

​Yes, because one would expect an intercept representing the variable cost of the ring.

D.

​No, because one would expect a positive​ intercept, representing the fixed cost of the ring.

​(b) Is

​ $650

an unusually high price for a ring with a diamond that weighs

0.25

​carat? Explain.

​(Round to the nearest integer as needed. Use ascending​ order.)

A.The price

is not

unusually high because it

is not

above the​ 95% prediction interval

$

to

$.

B.The price

is

unusually high because it

is

above the​ 95% prediction interval

$

to

$.

C.The price

is

unusually high because it

is not

above the​ 95% prediction interval

$

to

$.

D.The price

is not

unusually high because it

is

above the​ 95% prediction interval

$

to

$.

Weight (Carat)   Price ($)
0.29   836
0.21   536
0.23   649
0.23   566
0.29   791
0.19   469
0.24   636
0.26   753
0.22   581
0.17   368
0.23   558
0.17   423
0.29   788
0.19   454
0.23   650
0.18   407
0.26   728
0.28   839
0.17   413
0.22   603
0.25   699
0.28   835
0.27   800
0.15   274
0.18   449
0.22   599
0.18   454
0.23   574
0.27   762
0.18   457

Answer 1

Ho:   βo=   0  
H1:   βo╪   0  

estimated std error of intercept =Se(ßo) =    Se*√(1/n+x̅²/Sxx)=   31.1856
t stat = estimated intercept/std error =ßo /Se(ßo) =    (-236.1447-0)/31.1856=   -7.572
Degree of freedom ,df = n-2=   28  

      
p-value =    0.0000  
decison :    p-value<α , reject Ho  

A.Reject

H0.

There is sufficient evidence that the population intercept is not zero.

b)

No, because one would expect a positive intercept.

c)

X Value=   0.25              
Confidence Level=   95%              
                  
                  
Sample Size , n=   30              
Degrees of Freedom,df=n-2 =   28              
critical t Value=tα/2 =   2.048   [excel function: =t.inv.2t(α/2,df) ]          
                  
X̅ =    0.23              
Σ(x-x̅)² =Sxx   0.05              
Standard Error of the Estimate,Se=   30.9610              
                  
Predicted Y at X=   0.25   is          
Ŷ=   -236.14468   +   3703.45272   *0.25=   689.719
                  
                  
For Individual Response Y                  
standard error, S(ŷ)=Se*√(1+1/n+(X-X̅)²/Sxx) =   31.6513              
margin of error,E=t*std error=t*S(ŷ)=    2.048   *   31.651   =   64.8348
                  
Prediction Interval Lower Limit=Ŷ -E =   689.719   -   64.835   =   624.884
Prediction Interval Upper Limit=Ŷ +E =   689.719   +   64.835   =   754.553

A.The price is not unusually high because it is not above the​ 95% prediction interval 625 to 755