In: Statistics and Probability
A statistician took a walk down the street at noon during a day when schools were in session. She recorded the age and measured the height of the first children of different ages she encounter on this street. The oldest child was a six-year old.
The statistician made sure that all the children were unrelated to each other.
Here are the measurements
Child |
1 |
2 |
3 |
4 |
5 |
Age (years) |
2 |
3 |
4 |
5 |
6 |
Height (cm) |
87 |
90 |
102 |
112 |
110 |
SSx=10, SSy=516.8, SSxy=68, ∑x=20, ∑y=501, ∑xy=2072
Find the 95% confidence limits on the expected height of 2-year-old child on the street.
X | Y | XY | X² | Y² |
2 | 87 | 174 | 4 | 7569 |
3 | 90 | 270 | 9 | 8100 |
4 | 102 | 408 | 16 | 10404 |
5 | 112 | 560 | 25 | 12544 |
6 | 110 | 660 | 36 | 12100 |
Ʃx = | Ʃy = | Ʃxy = | Ʃx² = | Ʃy² = |
20 | 501 | 2072 | 90 | 50717 |
Sample size, n = | 5 |
x̅ = Ʃx/n = 20/5 = | 4 |
y̅ = Ʃy/n = 501/5 = | 100.2 |
SSxx = Ʃx² - (Ʃx)²/n = 90 - (20)²/5 = | 10 |
SSyy = Ʃy² - (Ʃy)²/n = 50717 - (501)²/5 = | 516.8 |
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 2072 - (20)(501)/5 = | 68 |
a)
Slope, b = SSxy/SSxx = 68/10 = 6.8
y-intercept, a = y̅ -b* x̅ = 100.2 - (6.8)*4 = 73
Regression equation :
ŷ = 73 + (6.8) x
b)
Null and alternative hypothesis:
Ho: β₁ = 0
Ha: β₁ ≠ 0
n = 5
α = 0.01
Slope, b = 6.8
Standard error of slope, se(b1) = 1.3466
Test statistic:
t = b/(s/√SSxx) = 6.8/(4.2/√10) = 5.12
df = n-2 = 3
p-value = T.DIST.2T(ABS(5.12), 3) = 0.0144
Conclusion:
p-value > α , Fail to reject the null hypothesis.
There is enough evidence to conclude that x is useful linear predictor of height change at 0.01 significance level.
--
Predicted value of y at x = 2
ŷ = 73 + (6.8) * 2 = 86.6
Critical value, t_c = T.INV.2T(0.05, 3) = 3.1824
95% confidence interval on the expected height of 2-year-old child on the street :
Lower limit = ŷ - tc*se*√(1 + (1/n) + ((x-x̅)²/(SSxx)))
= 86.6 - 3.1824*4.2*√(1 + (1/5) + ((2 - 4)²/(10))) = 69.69
Upper limit = ŷ + tc*se*√(1 + (1/n) + ((x-x̅)²/(SSxx)))
= 86.6 + 3.1824*4.2*√(1 + (1/5) + ((2 - 4)²/(10))) = 103.51