According to the Canadian Census, the proportion of families with three or more children (aged less than 25) declined from 42% in 1961 to 19% in 2011. Suppose we estimate that this proportion is 17.5% this year. In order to market products to large families, we randomly select 1600 Canadian families for a survey.

a) What is the expected proportion of families with three or more children in our sample?

b) In our sample, we find 19.3% of the families have three or more children. What is the probability of a proportion as high as 19.3% or higher in the sample if our estimate of 17.5% in the population is correct?

c) Comment on whether the survey result casts doubt on the population estimate.

d) Suppose we had surveyed a random sample of only 400 families instead of 1600 and had gotten the same result: 19.3% of the sample had three or more children. Comment on whether this result from a smaller survey casts doubt on the population estimate.