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.

Weight (Carat)	Price ($)
0.21	540
0.18	444
0.18	430
0.16	350
0.15	330
0.25	667
0.22	526
0.27	723
0.24	655
0.25	640
0.22	605
0.29	801
0.22	549
0.22	610
0.22	579
0.29	782
0.28	750
0.26	750
0.17	413
0.16	382
0.25	691
0.17	408
0.23	625
0.27	749
0.19	412
0.28	791
0.19	490
0.26	691
0.23	628
0.19	463

​(a) Could these data be a sample from a population in which the population intercept is​ zero? 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<0

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

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

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

D. ​No, because one would expect a negative​ intercept, representing the variable 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 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 not unusually high because it is notabove the​ 95% prediction interval [$ _____ to $ _____] .

D.The price is unusually high because it is not above the​ 95% prediction interval [$ ______ to $ _____] .

Answer 1

X	Y	XY	X²	Y²
0.21	540	113.4	0.0441	291600
0.18	444	79.92	0.0324	197136
0.18	430	77.4	0.0324	184900
0.16	350	56	0.0256	122500
0.15	330	49.5	0.0225	108900
0.25	667	166.75	0.0625	444889
0.22	526	115.72	0.0484	276676
0.27	723	195.21	0.0729	522729
0.24	655	157.2	0.0576	429025
0.25	640	160	0.0625	409600
0.22	605	133.1	0.0484	366025
0.29	801	232.29	0.0841	641601
0.22	549	120.78	0.0484	301401
0.22	610	134.2	0.0484	372100
0.22	579	127.38	0.0484	335241
0.29	782	226.78	0.0841	611524
0.28	750	210	0.0784	562500
0.26	750	195	0.0676	562500
0.17	413	70.21	0.0289	170569
0.16	382	61.12	0.0256	145924
0.25	691	172.75	0.0625	477481
0.17	408	69.36	0.0289	166464
0.23	625	143.75	0.0529	390625
0.27	749	202.23	0.0729	561001
0.19	412	78.28	0.0361	169744
0.28	791	221.48	0.0784	625681
0.19	490	93.1	0.0361	240100
0.26	691	179.66	0.0676	477481
0.23	628	144.44	0.0529	394384
0.19	463	87.97	0.0361	214369

Sample size, n =	30
Ʃ x =	6.7
Ʃ y =	17474
Ʃ xy =	4074.98
Ʃ x² =	1.5476
Ʃ y² =	10774670
x̅ =	0.223333333
y̅ =	582.4666667
SSxx = Ʃx² - (Ʃx)²/n =	0.051266667
SSyy = Ʃy² - (Ʃy)²/n =	596647.4667
SSxy = Ʃxy - (Ʃx)(Ʃy)/n =	172.4533333

Slope, b = SSxy/SSxx =    3363.849155          
y-intercept, a = y̅ -b* x̅ =    -168.7929779          
              
Regression equation :               
ŷ =     -168.7930   +   3363.8492   x

Sum of Square error, SSE = SSyy -b*Ssxy =    16540.4671
Standard error, se = √(SSE/(n-2)) =   24.30496

Standard error for slope, se(b1) = se/√SSxx =   107.3439

a) null and alternative hypotheses. Answer B

B. H0: β0=0​,
Ha: β0≠0

Test statistic:

p-value = T.DIST.2T(31.337, 28) = 0.0000

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

ANSWER C.Reject H0. There is sufficient evidence that the population intercept is not zero.

Should β0=​0?

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

b) Predicted value at X =   0.25          
ŷ =     -168.7930 + 3363.8492 * 0.25 =   672.1693

At α = 0.05 and df = n-2 = 28, critical value, t_c = T.INV.2T(0.05, 28 ) = 2.0484

95% prediction interval:

Answer C. The price is not unusually high because it is not above the​ 95% prediction interval [ $ 621.221 to $ 723.117 ] .