Chapter 6: Q. 6.14 (page 437)

According to the Census Bureau, $13 %$ of American adults (aged 18 and over) are Hispanic. An opinion poll plans to contact an SRS of $1200$ adults.
a. What is the mean number of Hispanics in such samples? What is the standard deviation?
b. Should we be suspicious if the sample selected for the opinion poll contains $10 %$ or less Hispanic people? Calculate an appropriate probability to support your answer.

Short Answer

Expert verified

The obtained values are $μ = 156$ and $σ = 11.6499$ .
Regardless of the fact that this likelihood is less than $0.05$ or $5 %$ , 180 Hispanics are unlikely to appear by chance, and hence suspicion should be raised.

Step by step solution

Part (a) Step 1: Given information

Given:

$\begin{array}{rcl} p & = & 13 % \\ = & 0.13 n \\ = & 1200 \end{array}$

Formula used:

$μ = n p σ = \sqrt{n p (1 - p)}$

Part (a) Step 2: Calculation

The average is

$\begin{array}{rcl} μ & = & n p \\ = & 1200 (0.13) \\ = & 156 \end{array}$

A binomial random variable's standard deviation is

\begin{array}{rcl} σ & = & \sqrt{n p (1 - p)} \\ = & \sqrt{1200 (0.13) (1 - 0.13)} \\ = & 11.6499 \end{array}

Part (b) Step 1: Given information

Given:

$15 %$ of the sample contains Hispanic.

\begin{array}{rcl} p & = & 13 % \\ = & 0.13 \end{array}

Formula used:

P (X = k) = ({}^{n}C_{k}) \times p^{k} \times (1 - p) n - k

Part(b) Step 2: Calculation

Because the likelihood of the number of successes in the sample is important, the distribution is binomial.

P (X = k) = ({}^{n}C_{k}) \times p^{k} \times (1 - p)^{n - k}

Determine if you should be suspicious or not by calculating the likelihood of receiving anything.

$1200 \times 15 % = 180$ Hispanics or more extreme:

P (X \geq 180) = \sum_{k = 180}^{1200} ({}^{n}C_{1200}) \times 0 . 13^{k} \times (1 - 0.13)^{1200 - k} = 0.0235

Despite the fact that this likelihood is less than $0.05$ or $5 %$ , 180 Hispanics are unlikely to appear by chance, and hence suspicion should be raised.

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Value of X	0	1	2	3	4
Probability	0.1	0.2	0.3	0.3	0.1

Value of Y	0	1	2	3
Probability	0.4	0.3	0.2	0.1

Value $(y_{i})$	0	1	2	3	4
Probability $(P_{i})$	$0.78$	$0.11$	$0.07$	$0.03$	$0.01$

Short Answer

Step by step solution

Part (a) Step 1: Given information

Part (a) Step 2: Calculation

Part (b) Step 1: Given information

Part(b) Step 2: Calculation

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