Question 1. In a city of 72,500 people, a simple random sample of four households is selected from the 25,000 households in a population to estimate the average cost on food per households for a week. The first household in the sample had 4 people and spent a total of $150 in food that week. The second household had 2 people and spent $100. The third, with 4 people, spent $200. The fourth, with 3 people, spent$140.

(a) Identify the sampling units, the variable of interest, and any auxiliary information associated with the units.

(b) Describe two types of estimators for estimating the mean expenditure per household for a week's food in the city.

(c) Estimate mean expenditure using the first estimator, and estimate the variance of the estimator.

(d) Estimate mean expenditure using the other estimator, and estimate the variance of the estimator.

(e) Based on the data, which estimator appears preferable in this situation?

Question 2. Suppose there are only two domains in a population, defined by indicator variable

           1 if unit i is in domain 1

X_i = 
           0 if unit i is in domian 2

Then, the two population domain means of variable Y are given by, respectively,

Y¯₁ = ∑^N_i=1 X_i Y_i/∑N_i=1X_i

and

Y¯₂ = ∑^N_i=1 (1 - X_i) Y_i/∑^N_i=1(1 - X_i)

If a simple random sample with replacement sample of size n is selected. Let y_i and x_i be the observations of Y and X of the selected i-th unit. Define the two estimators of Y¯₁ and Y¯₂ as

y¯₁ = ∑ⁿ_i=1x_iy_i/∑ⁿ_i=1x_i and y¯ = ∑ⁿ_i=1(1 - x )y /∑n (1 - x )

i=1

i=1 i

2 i=1

i i i=1 i

Show that the covariance of y¯1 and y¯2 is approximately equal to zero.