Chapter 10: Q 2. (page 639)

Literacy A researcher reports that $80 %$ of high school graduates, but only $40 %$ of high school dropouts, would pass a basic literacy test. Assume that the researcher’s claim is true. Suppose we give a basic literacy test to a random sample of $60$ high school graduates and a separate random sample of $75$ high school dropouts. ${\hat{p}}_{G}, {\hat{p}}_{D}$ be the sample proportions of graduates and dropouts, respectively, who pass the test.
a. What is the shape of the sampling distribution of ${\hat{p}}_{G} - {\hat{p}}_{D}$ . Why?
b. Find the mean of the sampling distribution.
c. Calculate and interpret the standard deviation of the sampling distribution.

Short Answer

Expert verified

a. The shape is approximately normal.

b. $μ_{{\hat{p}}_{G} - {\hat{p}}_{D}} = 0.40$

c. $σ_{{\hat{p}}_{G} - {\hat{p}}_{D}} = 0.07745$

Step by step solution

Given Information

It is given that $n_{G} = 60$

$n_{D} = 75$

$p_{G} = 0.80$

$p_{D} = 0.40$

Shape of p^G-p^D

Assuming shape of ${\hat{p}}_{G} - {\hat{p}}_{D}$ is normal.

Conditions are:

$n_{G} p_{G} \geq 10$

$n_{G} (1 - p_{G}) \geq 10$

$n_{D P D} \geq 10$

$n_{D} (1 - p_{D}) \geq 10$

n_{G} p_{G} = (60) (0.80) = 48

$n_{G} (1 - p_{G}) = (60) (1 - . 80) = (60) (. 20) = 12$

$n_{D} p_{D} = (75) (0.40) = 30$

$n_{D} (1 - p_{D}) = (75) (1 - 0.40) = (75) (0.60) = 45$

As all four conditions are satisfied, the shape of ${\hat{p}}_{C} - {\hat{p}}_{A}$ is approximately normal.

Mean of Sampling Distribution

Using $μ_{{\hat{p}}_{G} - {\hat{p}}_{D}} = p_{G} - p_{D}$

$= 0.80 - 0.40 = 0.40$

The mean is $0.40$

Standard Deviation

High School graduates, $(60) < 10 % of all high school graduates$

Dropouts, $(75) < 10 % of all high school graduates$

Formula as: $σ_{{\hat{p}}_{G} - {\hat{p}}_{D}} = \sqrt{\frac{p_{G} (1 - p_{G})}{n_{G}} + \frac{p_{D} (1 - p_{D})}{n_{D}}}$

$= \sqrt{\frac{0.80 (1 - 0.80)}{60} + \frac{0.40 (1 - 0.40)}{75}}$

$= \sqrt{\frac{0.80 (0.20)}{50} + \frac{0.40 (0.60)}{75}} \approx 0.07745$

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Short Answer

Step by step solution

Given Information

Shape of p^G-p^D

Mean of Sampling Distribution

Standard Deviation

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