Question

Please show all work AND any calculator functions.

The table below summarizes data of heights and weights of 8 randomly selected adults. Use this information to answer parts a) – h).

Height (x) in inches 5.2 5.6 5.8 5.9 5.4 6.1 6.0 5.7

Weight (y) in pounds 119 136 155 185 135 202 194 165

a) Test the claim ρ ≠ 0. Use α = 0.01.

• State the hypotheses and label the claim.

• State the significance level (α).

• Is the test left-tailed, right-tailed, or two-tailed (circle one)?

• What is the critical value(s)?

• What are the P-value and test-statistic?

• State the conclusion

b) Find the linear regression equation.

c) Graph the regression equation on a scatter plot.

d) Find the correlation coefficient. Make a conclusion about the strength of the correlation.

e) Give the coefficient of determination.

f) What percentage of the variation in y is explained by the variation in x?

g) What percentage of the variation in y is unexplained by the variation in x?

h) Predict y when x = 5.85

Given Answers to match= a) H0: ρ = 0 H1: ρ ≠ 0 (claim) α = 0.01; two-tailed; critical values: r = 0.834, t = ±3.707; test statistics: P-value = 0.0002, r = 0.9539, t = 7.786 Reject the null hypothesis. The P-value is less than α, the test statistic t is greater than 3.707 and falls in the critical region, and the test statistic r is greater than 0.834 and falls in the critical region. There is sufficient evidence to support the claim that ρ ≠ 0 and there is a linear correlation between height and weight. b) y = -382.4682 + 95.2023 d) r = 0.9539 strong positive correlation f) r2 = 0.9099 g) 90.99% explained h) 100% - 90.99% = 9.01% is unexplained i) y = 174.465

Answer 1

X	Y	XY	X²	Y²
5.2	119	618.8	27.04	14161
5.6	136	761.6	31.36	18496
5.8	155	899	33.64	24025
5.9	185	1091.5	34.81	34225
5.4	135	729	29.16	18225
6.1	202	1232.2	37.21	40804
6.0	194	1164	36	37636
5.7	165	940.5	32.49	27225

Ʃx =	45.7
Ʃy =	1291
Ʃxy =	7436.6
Ʃx² =	261.71
Ʃy² =	214797
Sample size, n =	8
x̅ = Ʃx/n = 45.7/8 =	5.7125
y̅ = Ʃy/n = 1291/8 =	161.375
SSxx = Ʃx² - (Ʃx)²/n = 261.71 - (45.7)²/8 =	0.64875
SSyy = Ʃy² - (Ʃy)²/n = 214797 - (1291)²/8 =	6461.875
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 7436.6 - (45.7)(1291)/8 =	61.7625

a) Null and alternative hypothesis:

Ho: ρ = 0 ; Ha: ρ ≠ 0

α = 0.01

Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 61.7625/√(0.64875*6461.875) = 0.9539

It is a two tailed test.

df = n-2 = 6

Critical value, t_c = T.INV.2T(0.01, 6) = 3.707

Test statistic :  

t = r*√(n-2)/√(1-r²) = 0.9539 *√(8 - 2)/√(1 - 0.9539²) = 7.786

p-value = T.DIST.2T(ABS(7.7861), 6) = 0.0002

Conclusion:

p-value < α Reject the null hypothesis. There is a correlation between x and y.

b)

Slope, b = SSxy/SSxx = 61.7625/0.64875 = 95.202312

y-intercept, a = y̅ -b* x̅ = 161.375 - (95.20231)*5.7125 = -382.4682

Regression equation :

ŷ = -382.4682 + (95.2023) x

c) Graph:

d) Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 61.7625/√(0.64875*6461.875) = 0.9539

It has a strong positive correlation

e) Coefficient of determination, r² = (SSxy)²/(SSxx*SSyy) = (61.7625)²/(0.64875*6461.875) = 0.9099

f) Explained = 90.99%

g) Unexplained = 9.01%

h) Predicted value of y at x = 5.85

ŷ = -382.4682 + (95.2023) * 5.85 = 174.4653