Question

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

a) Find the equation describing the linear relationship between age and height for 2– to 6-year old children on that street.

b) Conduct a test to determine if age, x, is useful linear predictor of height change. Assume s=4.2 and use a = 0.01.

c) Find the 95% confidence limits on the expected height of 3.5 year-old child on the street.

Answer 1

using excel>addin>phstat>simple regression '

we have

Simple Linear Regression Analysis
Regression Statistics
Multiple R	0.9459
R Square	0.8947
Adjusted R Square	0.8596
Standard Error	4.2583
Observations	5
ANOVA
	df	SS	MS	F	Significance F
Regression	1	462.4000	462.4000	25.5000	0.0150
Residual	3	54.4000	18.1333
Total	4	516.8000
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	73.0000	5.7131	12.7776	0.0010	54.8182	91.1818
Age	6.8000	1.3466	5.0498	0.0150	2.5145	11.0855

Confidence Interval Estimate
Data
X Value	3.5
Confidence Level	95%
Intermediate Calculations
Sample Size	5
Degrees of Freedom	3
t Value	3.182446
XBar, Sample Mean of X	4
Sum of Squared Differences from XBar	10
Standard Error of the Estimate	4.258325
h Statistic	0.225
Predicted Y (YHat)	96.8
For Average Y
Interval Half Width	6.4282
Confidence Interval Lower Limit	90.3718
Confidence Interval Upper Limit	103.2282
For Individual Response Y
Interval Half Width	14.9992
Prediction Interval Lower Limit	81.8008
Prediction Interval Upper Limit	111.7992

a) the equation describing the linear relationship between age and height for 2– to 6-year old children on that street.

height = 73 + 6.8 Age

b) we will use t test , from output t stat= 5.0498

p value is 0.015

since p value is greater than 0.01 so we conclude that age is not a good predictior.

c) the 95% confidence limits on the expected height of 3.5 year-old child on the street is 81.0008,111.7992