2. The data set infmort from the faraway package contains information on the mor

Question

2. The data set infmort from the faraway package contains information on the mortality of infants for 105 nations. The variable mortality gives the number of deaths per 1000 live births, while income is the per capita income in US dollars and region indicates the geographic area of the nation Suppose that the response Y = log(mortality), one predictor 11 = log(income), and another predictor z2 = region. Consider the model (a) State the null and alternative hypotheses for the overall F-test for this model. Perform the test (b) Explain the practical meaning of the hypothesis Ho : (c) Perform a test for the hypothesis in part (b) and summarize your results and summarize results. 2 = 0 in the context of the above = model.

Explanation / Answer

(b)

The practical meaning of the hypothesis H0 : 12 = 2 = 0 is that for given income and region, only log(income) is significant in determining log(mortality). In other words, log(mortality) is independent of region and interaction between regio and log(income) for given income and region.

(c)

The unrestricted model is

E(log(mortality)|income, region) = 0 + 1 log(income) + 2region + 12region log(income)

R Code to run the above model is -

model.ur = lm(log(mortality) ~ log(income) + region + region*log(income), data = infmort)

The restricted model is

E(log(mortality)|income, region) = 0 + 1 log(income)

R Code to run the above model is -

model.r = lm(log(mortality) ~ log(income), data = infmort)

R-Squared for both the models are

> summary(model.r)$r.squared
[1] 0.5021236
> summary(model.ur)$r.squared
[1] 0.6464492

F -Statisitic is given as,

F = [(R2UR - R2R)/q] / [(1-R2UR) / (n-k)]

where R2UR and R2R are R-squared values for unrestricted and restricted model.

q is number of restrivtions. Here q = 2

n is number of observations. n = 105

k is number of coefficients (including intercept) in the unrestricted model . Here k = 4

F = [(0.6464492 - 0.5021236)/2] / [(1-0.6464492) / (105-4)] = 20.61498

Degree of freedom = q , n-k = 2, 101

Critical value of F for significance level of 0.05 and df = 2, 101 is 3.086

As, observed value of F is greater than the criticacl value, we reject the null hypothesis

As, we reject the null hypothesis, we conclude that at 5% significance level, 12 and 2 are not 0 and region and interaction between region and log(income) are significant variable in determining the (log(mortality) for given income and region.

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