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Svetach [21]
3 years ago
13

In a beach town, 13% of the residents own boats. A random sample of 100 residents was selected. What is the probability that les

s than 11% of the residents in the sample own boats?
Find the z-table here.
Mathematics
2 answers:
Svetlanka [38]3 years ago
8 0

Answer:

0.276

Step-by-step explanation:

got it right on edge

mariarad [96]3 years ago
4 0

Answer:

Approximately 0.038 (or equivalently 3.8\%,) 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 n = 100 selected residents owns boats as a Bernoulli random variable: for k = \lbrace 1,\, \dots,\, 100\rbrace, X_k \sim \text{Bernoulli}(\underbrace{0.13}_{p}).

X_k = 0 means that the kth resident in this sample does not own boats. On the other hand, X_k = 1 means that this resident owns boats. Therefore, the sum (X_1 + \cdots + X_{100}) would represent the number of residents in this sample that own boats.

Each of these 100 random variables are all independent from one another. The mean of each X_k would be \mu = 0.13, whereas the variance of each X_k\! would be \sigma = p\, (1 - p) = 0.13 \times (1 - 0.13) = 0.1131.

The sample size of 100 is a rather large number. Besides, all these samples share the same probability distribution. Apply the Central Limit Theorem. By this theorem, the sum (X_1 + \cdots + X_{100}) would approximately follow a normal distribution with:

  • mean n\, \mu = 100 \times 0.13 = 13, and
  • variance \sigma\, \sqrt{n} = p\, (1 - p)\, \sqrt{n} = 0.1131 \times 10 = 1.131.

11\% of that sample of 100 residents would correspond to 11\% \times 100 = 11 residents. Calculate the z-score corresponding to a sum of 11:

\begin{aligned}z &= \frac{11 - 13}{1.131} \approx -1.77 \end{aligned}.

The question is (equivalently) asking for P( (X_1 + \cdots + X_{100}) < 11). That is equal to P(Z < -1.77). However, some z-tables list only probabilities like P(Z > z). Hence, convert P(Z < -1.77)\! to that form:

\begin{aligned} & P( (X_1 + \cdots + X_{100}) < 11) \\ &= P(Z < -1.77) \\ &= 1 - P(Z >1.77) \end{aligned}.

Look up the value of P(Z > 1.77) on a z-table:

P(Z > 1.77) \approx 0.962.

Therefore:

\begin{aligned} & P( (X_1 + \cdots + X_{100}) < 11) \\ &= 1 - P(Z >1.77) \\ &\approx 1 - 0.962 = 0.038 \end{aligned}.

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