Suppose that 20% of the customers of a cable television company watch the shopping channel at least once a week. The cable company does not know the actual proportion of all customers who watch the shopping channel at least once a week and is trying to decide whether to replace this channel with a new local station. The company plans to take a random sample of 100 customers and to use ˆ p as an estimate of the population proportion.
a. Show that σ ˆ p, the standard deviation of ˆ p , is equal to 0.0400.
b. If for a different sample size, σ ˆ = 0.0231, would you expect more or less sample- to- sample variability in the sample proportions than when n = 100?
c. Is the sample size that resulted in σ ˆ = 0.0231 larger than 100 or smaller than 100? Explain your reasoning.