Consider the following problem statement:

An article in an online magazine states that 40% of home buyers found their real estate agent through referrals by a friend. However, a professor in a local college sampled 1000 home buyers and found that 426 chose an agent recommended by a friend.

Does the data refute the claim made by the magazine? Use a significance level of 0.02.

Solution:

• First, you should recognize that this is a test about a single proportion, not a mean or other statistic.
• The claim is that the proportion of home buyers who select their real estate agent based on the recommendation of a friend is 0.40. Therefore, the claim is p = 0.40.
• Since the claim contains an equality, =, it must be the null. Ho: p = 0.40.
• The alternative must be the complement, Ha: p ≠ 40.
• Remember the rule of thumb is that all hypothesis tests for proportions are z-tests. But you should confirm that you can use the normal distribution by checking that both n*p and n*q are greater than 5. Here n*p = 2000*0.40 = 800 and n*q = 2000*(1-0.40) = 1200. Both are > 5, there we can use the normal distribution.
• I recommend always sketching the situation described in the problem. Here we see that the sample count of 426 falls on the right side of the hypothesized mean of 400 for the population. Recall, the mean for a proportion is just the n*p or 0.4 * 1000. The standard deviation for a proportion is

Single-sample Hypothesis Test for a Proportion