Although the department store chain in Example 8.12 has randomly selected 1 percent of the homes in each stratum, there is a more precise way to determine how to sample the strata. The first step is to determine the optimal fraction, f_i, of the total sample size, n, that will be allocated to the ith stratum. This optimal fraction can be proven to give the shortest confidence intervals for a fixed sample size and can be proven to equalHere, c_i is the cost of sampling the ith stratum, and σ_i is the true standard deviation of the measurements in the ith stratum. If the sampling costs are equal (as they might be when conducting telephone interviews)In the department store example, we will assume that the sampling costs are equal, and we will estimate σ₁,σ₂, and σ₃ by s₁,s₂, and s₃.We can then estimate N₁σ₁,N₂σ₂, and N₃σ₃, to be N₁S₁ = 37,108,776, N₂s₂ = 34,904,638, and N₃s₃ = 13,602,042. Here, we have used the N_i and s_i values in Table 8.4 on page 330. The sum (N₁s₁+ N₂s₂ + N₃s₃) is 85,615,456. 

Therefore, the optimal sampling fractions are:

f₁=37,108,77685,615,456 = .4334,

f₂=34,904,63885,615,456 = .4077,

f₃=13,602,04285,615,456 = .1589.

Next, consider the sample size formula

Data from Example 8.12

The department store chain has concluded that it should open the new store in the geographical region if

(1) it is 95 percent confident that the mean income per household is at least $50,000 and

(2) it is 95 percent confident that the total number of households for which at least one household member would be likely to shop at the new store is at least 10,000. Using the information in Table 8.4, a point estimate of the total income,τ, of all households in the geographical region is(dollars). To compute a 95 percent confidence interval for t, we first compute the margin of error in this interval, which is

Transcribed Image Text:

fi = No;/Vc; ₁0₁/√₂ + N₂0₂/√©₂ + ... + N₂0₂ / √CK