Chapter 7: Q.7.21 (page 296)

America's Riches. Each year, Forbes magazine publishes a list of the richest people in the United States. As of September l6, 2013, the six richest Americans and their wealth (to the neatest billion dollars) are as shown in the following table. Consider these six people a population of interest.
(a) For sample size of $5$ construct a table similar to table 7.2 on page293.(There are 6 possible sample) of size $5$
(b) For a random sample of size $5$ determine the probability that themean wealth of the two people obtained will be within $3$ (i.e, $3$ billion) of the population mean. interpret your result in terms of percentages.

Expert verified

(a)

(b) The probability that $x$ is within 3 billon of $μ i s 0.8333$

Step by step solution

Given in the question that,

we have to construct a table with sample size of $5$

The sample mean of size $5$ :

$\bar{x} = \frac{\sum_{i = 1}^{5} x_{i}}{5}$

Thus, the samples of size $5$ and the corresponding means are obtained as shown in the table below:

Given in the question that,

we have to determine the probability that the mean wealth of the two people obtained will be within $3$ (i.e, $3$ billion) of the population mean.

We have to find $P (μ - 3 \leq \bar{x} \leq μ + 3)$ $P (μ - 3 \leq \bar{x} \leq μ + 3)$

$P (μ - 3 \leq \bar{x} \leq μ + 3) = P (46.5 - 3 \leq \bar{x} \leq 46.5 + 3)$

$= P (43.5 \leq \bar{x} \leq 49.5)$

$= \frac{5}{6} = 0.8333$

Therefore, the probability that $x$ is within $3$ billion of $μ i s 0.8333$

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