Question

Automobile racing, high-performance driving schools, and driver education programs run by automobile clubs continue to grow in popularity. All these activities require the participant to wear a helmet that is certified by the Snell Memorial Foundation, a not-for-profit organization dedicated to research, education, testing, and development of helmet safety standards. Snell "SA" (Sports Application) rated professional helmets are designed for auto racing and provide extreme impact resistance and high fire protection. One of the key factors in selecting a helmet is weight, since lower weight helmets tend to place less stress on the neck. The following data show the weight and price for 18 SA helmets. Helmet Weight (oz) Price ($) 245 Pyrotect Pro Airflow Pyrotect Pro Airflow Graphics RCi Full Face Race Quip RidgeLine HJC AR-10 HJC Si-12 HJC HX-10 Impact Racing Super Sport Zamp FSA-1 Zamp RZ-2 Zamp RZ-2 Ferrari 304 Zamp RZ-3 Sport Zamp RZ-3 Sport Painted Bell M2 Bell M4 Bell M4 Pro G Force Pro Force 1 G Force Pro Force 1 Grafx a. Develop a scatter diagram with weight as the independent variable. Choose the correct a scatter diagram:

Answer 1

Weight(X)	Price(Y)	XY	X²	Y²
64	245	15680	4096	60025
64	275	17600	4096	75625
64	205	13120	4096	42025
64	201	12864	4096	40401
58	303	17574	3364	91809
47	709	33323	2209	502681
49	900	44100	2401	810000
59	331	19529	3481	109561
66	204	13464	4356	41616
58	308	17864	3364	94864
58	304	17632	3364	92416
52	489	25428	2704	239121
52	485	25220	2704	235225
63	361	22743	3969	130321
62	370	22940	3844	136900
54	565	30510	2916	319225
63	251	15813	3969	63001
63	279	17577	3969	77841
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
1060	6785	382981	62998	3162657

Sample size, n =	18
x̅ = Ʃx/n = 1060/18 =	58.8888889
y̅ = Ʃy/n = 6785/18 =	376.944444
SSxx = Ʃx² - (Ʃx)²/n = 62998 - (1060)²/18 =	575.777778
SSyy = Ʃy² - (Ʃy)²/n = 3162657 - (6785)²/18 =	605088.944
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 382981 - (1060)(6785)/18 =	-16580.111

Sum of Square error, SSE = SSyy -SSxy²/SSxx = 605088.94444 - (-16580.11111)²/575.77778 = 127647.66

Standard error, se = √(SSE/(n-2)) = √(127647.65535/(18-2)) = 89.31953

a) Scatter plot: Answer B.

B. Price $ 55 Weight 02

b) There appears to be a negative linear relationship between variables.

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c) Slope, b = SSxy/SSxx = -16580.11111/575.77778 = -28.79602

y-intercept, a = y̅ -b* x̅ = 376.94444 - (-28.79602)*58.88889 = 2072.7103

Regression equation :

Price = 2072.7 + (-28.8) Weight

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For Intercept :

y-intercept, a = y̅ -b* x̅ = 376.94444 - (-28.79602)*58.88889 = 2072.71

Standard error for Intercept, se(b0) = se*√((1/n) + (x̅²/SSxx)) = 89.31953*√((1/18) + (58.88889²/575.77778)) = 220.2146

t = a /se(b0) = 9.412229

For slope :

Slope, b = SSxy/SSxx = -16580.11111/575.77778 = -28.796

Standard error for slope, se(b1) = se/√SSxx = 89.31953/√575.77778 = 3.7223652

t = b /se(b1) = -7.73595

	Coefficients	Standard Error	t Stat
Constant	2072.7	220.215	9.412
Weight	-28.8	3.722	-7.736

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S = se = 89.3196

R Square = (SSxy)²/(SSxx*SSyy)

= (-16580.11111)²/(575.77778*605088.94444) = 0.7890 = 78.90%

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df(Regression) = 1

df(residual) = n-2 = 16

df(total) = n-1 = 17

SSR = SSxy²/SSxx = 477441.29

SSE = SSyy - SSxy²/SSxx =127647.66

SST = SSyy = Ʃy² - (Ʃy)²/n =605088.94

MSR = SSR/df(regression) = 477441.29

MSE = SSE/df(residual) = 7977.9785

F = MSR/MSE = 59.844896

p-value = F.DIST.RT(59.8449, 1, 16) = 0.0000

ANOVA
	df	SS	MS	F	Significance F
Regression	1	477441.3	477441.3	59.84	0.0000
Residual	16	127647.7	7978.0
Total	17	605088.9

p-value is less than 0.01

d) We can conclude that their is a significant relationship between variables.

e) Correlation coefficient, r = SSxy/√(SSxx*SSyy)

= -16580.11111/√(575.77778*605088.94444) = -0.8883