Researchers would like to know whether the proportions of elementary school chil

Question

Researchers would like to know whether the proportions of elementary school children who are obsess differ in rural and urban areas. Random samples of 900 children from each type of area are taken. Suppose that in fact 20% of the children are obese in both populations. ( I figured the answers to (a), (b), and (c). But I don't know how to solve (d), why is 0.05 plausible? plz explain this part)

(a) Give numerical values for the mean and standard deviation of the sampling distribution of ˆp 1ˆp 2.

(b) Suppose that the samples are taken and the difference in sample proportions is 0.05. Find the standardized statistic corresponding to this difference.

(c) Draw a picture of the sampling distribution, and illustrate where the difference of 0.05 falls on it.

(d) If the population proportions are really equal, is a difference in sample proportions of 0.05 plausible?

Explanation / Answer

d) Let us say the population proportions are really equal, i.e the number of obese ppl in rural and urban are exactly the same. 200 in 10000 of rural and 200 in 10000 of urban.

When we are taking random samples of 900 each in both the groups, it is possible that we get the count of obese as

180 in rural, 180 in urban

150 in rural, 200 in urban

200 in rural, 150 in urban

200 in rural, 200 in urban

240 in rural, 125 in urban

So there are different possibilities, but mostly it should be 180 each in both the sample groups as the population proportion in 20%. How do you define "mostly"? That is where the alpha or the leeway of 0.05 comes in to play. if the difference is 5% say urban to rural or viceversa is 25% and 20% or 15% and 13% or 30% and 32% we ignore that and attribute that difference in proportions as non-significant.

But we cannot ignore the fact it is plausible that the difference can be 0.05

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