Question

3. In the process of collecting weight and height data from 29 female and 81 male students at your uni- versity, you also asked the students for the number of siblings they have. Although it was not quite clear to you initially what you would use that variable for, you construct a new theory that suggests that children who have more siblings come from poorer families and will have to share the food on the table. Although a friend tells you that this theory does not pass the "straight-face" test, you decide to hypothesize that peers with many siblings will weigh less, on average, for a given height. In addition, you believe that the muscle/fat tissue composition of male bodies suggests that females will weigh less, on average, for a given height. To test these theories, you perform the following regression Studentw =-229.2 - 6.52 x Female + 0.51 x Sibs + 5.58 x Height, (5.52) (0.62) (44.01) R2 0.5, SER = 21.08 (2.25) where Studentw is in pounds, Height is in inches, Female takes a value of 1 for females and is 0 otherwise, Sibs is the number of siblings (heteroskedasticity-robust standard errors in parentheses) (a) Carrying out hypotheses tests using the relevant t-statistics to test your two claims separately, is there strong evidence in favor of your hypotheses? Is it appropriate to use two separate tests in this situation? (b) You also perform an F-test on the joint hypothesis that the two coefficients for females and siblings are zero. The calculated F-statistic is 0.84. Find the critical value from the F-table. Can you reject the null hypothesis? Is it possible that one of the two parameters is zero in the population, but not the other? (c) You are now a bit worried that the entire regression does not make sense and therefore also test for the height coefficient to be zero. The resulting F-statistic is 57.25. Does that prove that there is a relationship between weight and height?

Answer 1

ANSWER:
(a)

Claim 1:

H₀: Coefficient of Female = 0

H₁: Coefficient of Female $\neq$ 0

Test Statistic, t = -6.52 / 5.52 = -1.8

Degree of freedom = n - k - 1 where n is number of observations, k is number of predictors

= (29 + 81) - 3 - 1 = 106

Critical value of t at significance level of 0.05 and df = 106 is,$\pm$1.98

As, observed t is not less than -1.98 or greater than 1.98, we fail to reject H₀ and conclude that there is no significant evidence that the variable Female is significant in the model.

H₀: Coefficient of Sibs = 0

H₁: Coefficient of Sibs $\neq$ 0

Test Statistic, t = 0.51/ 2.25 = 0.2267

As, observed t is not less than -1.98 or greater than 1.98, we fail to reject H₀ and conclude that there is no significant evidence that the variable Sibs is significant in the model.

For testing simultaneously, number of hypothesis test is 2.

H₀: Coefficient of Female = 0

H₁: Coefficient of Sibs = 0

Using Boneferroni correction, Significance level = 0.05/2 = 0.025

Critical value of t at significance level of 0.025 and df = 106 is, $\pm$ 2.27

As, observed t for both the hypotheses is not less than -2.27 or greater than 2.27, we fail to reject H₀ and conclude that there is no significant evidence that the variable female and Sibs are significant in the model.

(b)

For F test,

Numerator degree of freedom = Number of variables in reduced model , k = 2

Denominator degree of freedom = n - k - 1= (29 + 81) - 2 - 1 = 107

Critical value of F for significance level of 0.05 and DF = 2, 107 is 3.08

As, observed F (0.84) is less than the critical value, we fail to reject H₀ and conclude that there is no significant evidence that the variable Sibs or Female is significant in the model.

(c)

For Full model,

Numerator degree of freedom = Number of variables in full model , k = 3

Denominator degree of freedom = n - k - 1= (29 + 81) - 3 - 1 = 106

Critical value of F for significance level of 0.05 and DF = 3, 106 is 2.69

As, observed F (57.25) is less than the critical value, reject H₀ and conclude that there is significant evidence that the model is significant and there is significant relationship between weight and height.

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