Question

# The frame for an education survey includes 2000 high schools, each of which contain 1000 students....

The frame for an education survey includes 2000 high schools, each of which contain 1000 students. A sample of n=3000 students is selected in two stages. At the first stage, 100 school are randomly selected. In the second stage, 30 students are randomly selected at each school. Of the selected students, 60% reported having access to a computer at home. A published estimate gives the standard error of this percentage as 1.4%. Ignoring the finite population correction and approximating (n-1) by n, use the sample proportion to estate the variance for a SRS of 3000 students. Use this value to estimate a design effect for this survey.

I think we need to use the equation deff=Var(estimator from design)/Var(estimator from SRS) but not sure how to set it up.

Given that n=3000

Sample proportion of students having access to computer at home p = 60% = 60/100 = 0.6

A published estimate gives the standard error of this percentage as 1.4% = 1.4/100 = 0.014 which is taken as the standard error of estimate of percentage of the design.

Therefore , Var(estimate of percentage of the design) = Square of standard error = (0.014)2 = 0.000196

Now, Var(estimate of percentage under SRS) = p(1-p)/n by ignoring finite population correction and replacing n-1 by n

= 0.6(1-0.6)/3000 = 0.6x0.4/3000 = 0.00008

Therefore, the design effect = deff = Var(estimate of percentage of the design)/ Var(estimate of percentage under SRS)

= 0.000196/0.00008 = 2.45

A deff of 2.45 means the variance is 2.45 times as large as the cvariance expected with SRS.

For cluster sampling, the sample size may be taken 2.45 times the sample size with SRS.

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