1. Suppose we have a random sample of 50 people and their weight, W and height,

Question

1. Suppose we have a random sample of 50 people and their weight, W and height, H are recorded

to the nearest pound and inch respectively. A regression of W on H and an intercept

gives:

Wi = 99.41(6.45) + 3.94(1.86)Hi

R^2 = 0.81, SER =      (sqrt{Su^2})     = 10.1 S= 10.1

(Standard errors are given in parentheses)

(h). Prove that if Wi* =      (lambda1+lambda2Wi)     then the new intercept b1* =      (mu1+mu2b1)      where b1 is the intercept of the original regression of W on H.

(k). Prove that if Wi* =      (lambda1+lambda2Wi)      and Hi* =       (mu1+mu2Hi)      

then the new slope is b2* =      ((lambda2/mu2)b2)      where b2 is the slope of the original regression of W on H.

(m). Explain how the OLS estimator of the variance of the disturbance term in (k) is related to

that in (h).

(n). Explain how the standard error of the slope coefficient in (k) is related to that in (h).

Explanation / Answer

n = 50

SER = 10.1

b2 = 3.94

SE(b2) = 1.86

Ho: b2=0

t = [(b2-0)/SE(b2)] = 3.94/1.86 = 2.1183

two tailed test,

at 5% CI, t(.025,50) = 2.009< 2.1183, hence hypothesis rejected

at 1% CI, t(.005,50) = 2.678>2.1183, hence hypothesis not rejected

one tailed test,

at 5% CI, t(.05,50) = 1.676< 2.1183, hence hypothesis rejected

at 1% CI, t(.01,50) = 2.403>2.1183, hence hypothesis not rejected

interpretation:

hypothesis rejected implies that weight is dependent upon height

hypothesis not rejected implies that weight is not dependent upon height

One sided t-test is justified on the assumption that b2=0 or b2>0 i.e we expect students with greater height to have greater weight and not otherwise.

Type I error: Occurs when the null hypothesis (Ho) is true, but is rejected

Type II error: Occurs when the null hypothesis is false, but erroneously fails to be rejected

The value of alpha, which is related to the level of significance(1-alpha) is directly related to type I errors. Alpha is the maximum probability that we have a type I error.

for two tailed test, alpha=5% and 1%

for one tailed test, alpha = 10% and 2%

A test statistic is said to be significant only if it lies in the region of alpha, which is double in the case of one tailed test. only in this region is the hyppothesis rejected. Therefore, if the Type I error is reduced, Type II error increases.

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