In a beach town, 13% of the residents own boats. A random sample of 100 residents was selected. What is the probability that less than 11% of the residents in the sample own boats? Find the z-table here.

Approximately (or equivalently ,) assuming that whether each resident owns boats is independent from one another.

Step-by-step explanation:

Assume that whether each resident of this town owns boats is independent from one another. It would be possible to model whether each of the selected residents owns boats as a Bernoulli random variable: for , .

means that the th resident in this sample does not own boats. On the other hand, means that this resident owns boats. Therefore, the sum would represent the number of residents in this sample that own boats.

Each of these random variables are all independent from one another. The mean of each would be , whereas the variance of each would be .

The sample size of is a rather large number. Besides, all these samples share the same probability distribution. Apply the Central Limit Theorem. By this theorem, the sum would approximately follow a normal distribution with:

of that sample of residents would correspond to residents. Calculate the -score corresponding to a sum of :

.

The question is (equivalently) asking for . That is equal to . However, some -tables list only probabilities like . Hence, convert to that form:

.

Look up the value of on a -table:

.

Therefore:

.

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