Question

1. A sociologist is interested in estimating the proportion of all parents that approve of “spanking” children when they deem it appropriate. Since this would seem to be a sensitive topic, the sociologist constructs a deck of cards that contains 75 cards marked “A” (approve of spanking) and 25 cards marked “D” (disapprove of spanking). A sample of 400 parents is selected; each is asked to draw a card from the deck and respond “Yes” if the letter on the card matches their opinion concerning the spanking of children. The experiment yields 120 “yes” responses. Estimate the proportion of parents that approve of spanking children, providing an appropriate bound on your error in estimation and a sentence interpreting your result.

Answer 1

1. Binomial "exact" calculation:

Proportion of positive results = P = x/N = 0.300

Lower bound = 0.255

Upper bound = 0.348

2. Normal approximation to the binomial calculation:

Standard error of the mean = SEM = √x(N-x)/N³ = 0.023

α = (1-CL)/2 = 0.025

Standard normal deviate for α = Z_α = 1.960

Proportion of positive results = P = x/N = 0.300

Appropriate bound

Lower bound = P - (Z_α*SEM) = 0.255

Upper bound = P + (Z_α*SEM) = 0.345

We are 95% confident that in a population parents spanking their children will be between 0.255 to 0.345 or 25.5% to 34.5%