Let the regression model that we are estimating be
where y=height (cm)
x=Age (years)
We know the following
a) An estimate of the slope is
An estimate of the intercept is
ans: The equation describing the linear relationship between age (x) and height (y) for 2-6 year old children on the street is
b) Age would be a useful linear predictor of height change if
the slope coefficient
is not equal to zero.
The hypotheses are
We know following
is the estimate of slope
The estimated standard error of slope is
The hypothesized value of the slope coefficient is
The test statistic is
Note: We have used s=4.2 in the above calculations. you would get t=5.049 only if you use an unrounded value of s.
This is a 2 tailed test (The alternative hypothesis has "not equal to")
The right tail critical value for
is
The degrees of freedom are n-2=5-2=3
Using the t tables, for df=3, and the area under the right
tail=0.005, we get
The critical values are -5.841, 5.841 (Note: 2 tail tests always have 2 critical values)
We will reject the null hypothesis if the test statistic does not lie within the acceptance region -5.841 to 5.841
Here, the test statistic is 5.120 and it lies within -5.841 to 5.841. Hence we do not reject the null hypothesis.
ans: We can conclude that there is no sufficient evidence to support the claim that age,x is useful linear predictor of height change.
c) The expected value of y (height) for x=A=4.5 is
The standard error of expected value of y for x=4.5 is
The significance level for 95% confidence level is
The right tail critical value is
Uing the t tables for df=3 and the area under the right
tail=0.025, we get
The 95% confidence interval is
ans: The 95% confidene interval on the expected height of a 4.5 year old child on the street is (97.2607, 109.9393) cm