Question

Models with quadratic terms: Using data from a random sample of 305 women, we have estimated the following model that relates a woman's sleep time in a week (in minutes) to her work time (in minutes) and age. SLEEP4206.17 -0.13 WRK- 37.64 AGE+0.47 AGE (333.60)(0.03) (17.46) (0.21) n - 305, R-0.092 (a) Explain the insights that the regression results provide for the effect of age on sleep, all else equal. In particular, all else equal, at what age women are predicted to sleep the least on average according to this estimated equation? (b) Explain how you would test each of the following two hypotheses (no need to perform the test, only need to state the null and the alternative, the test statistics and its distribution under the null, and if needed, the regression that need to be estimated to calculate the test statistic) i. Expected value of sleep time conditional on work time and age is a linear function of age. ii. Given work time, age is not a significant predictor of sleep time.

Answer 1

a) To calculate at what age women sleep the least on average, find a derivative of sleep with respect age and equal it to zero. Then, solve for the age. These steps are performed below:

S = 4206. 17-0.13W-37.64.4 + 0.47A2

Where sleep is denoted by , age is denoted by and work is denoted by .

Taking derivative of the above equation with respect to gives

dS =-37.64 0.47 * 2 * A-0 .A

Adding 37.64 on both sides of the above equation gives

0.47 2 * A -37.64

Dividing by 0.94 on both sides of the above equation gives

A-37.64/0.94-40.04

So, the age at which women work the least is 40.04 years.

Now, we can interpret the regression equation as follows:

The amount of sleep women get decreases with Age till 40.04 years and then increases after 40.04 years.

b) i) The value to Sleep time can be computed as

$S = a+ bW+ cA+ dA^2+ \epsilon '$

Now, to test if the expected value of sleep time conditional on work time and age is a linear function of age, we need to test the coefficient of $A^2$.

So, hypothesis test can be formulated as:

Ho : d = 0

$H_1 : d \neq 0$

where are the null and alternative hypothesis respectively.

Under the null hypothesis,

$S = a+bW+cA+\epsilon$

Test statistic will be

T=

test statistic follows F distribution with degrees of freedom.

ii) In this case, null and alternative hypothesis will be given as follows:

$H_1 : d \neq 0 | c \neq 0$

Under null hypothesis,

$S = a+bW+\epsilon$

Test statistic will be

T=

test statistic follows F distribution with degrees of freedom.