Two independent surveys were conducted: one in 1996 and one in 2006, on samples

Question

Two independent surveys were conducted: one in 1996 and one in 2006, on samples of 1,673 households from a national population. One of the questions asked was about the number of mobile telephones in each household. In the first survey, it was found that the mean number of mobiles per household was 0.59. In the second survey, the mean number was 1.47. The variances on the two occasions were 0.59 and 0.89, respectively. a. What is the difference in the mean number of mobiles and what is the variance of that difference? b. Suppose, now, that the samples were of the same households on both occasions, which means that the samples are not independent. Assume that the covariance between the two samples is 0.25. What is the variance of the difference in the number of mobiles between the two occasions?

Explanation / Answer

sample size= n= 1673

first survey- mean1 =0.59, var=0.59

second survey- mean 2= 1.47, var=0.89

since both the surveys were done independently we use t test for independent samples, thus

diff in means is= 1.47-0.59=0.88,

var of that diff= V(mean1-mean2)= var1/n + var2/n= 0.59/1673 +0.89/1673= 0.5905

second case when the sample are no longer independent

var(mean1-mean2)= var1+var2 - 2cov= 0.59+0.89-2*0.25=0.98

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