Chapter 11: Q. 11.69 (page 462)

In this Exercise, we have given the number of successes and the sample size for a simple random sample from a population. In each case,
a. use the one-proportion plus-four z-interval procedure to find the required confidence interval.
b. compare your result with the corresponding confidence interval found in Exercise $11.25 - 11.30$ , if finding such a confidence interval was appropriate.
role="math" localid="1651325715651" $x = 8, n = 40, 95 % level$

Short Answer

Expert verified

(a) The one-proportion $z$ -interval technique is adequate because $x$ and $n - x$ are both $5$ or more.

(b) It is possible to be $95 %$ convinced that the confidence interval is between $0.076$ and $0.324$ .

Step by step solution

Part(a) Step 1: Given Information

The size of a simple random sample from a population, as well as the number of successes.

$x = 8, n = 40, 95 % level$

$∴ n - x = 40 - 8 = 32$ , here $x$ and $n - x$ are both $5$ or greater.

Part(a) Step 2: Explanation

The sample proportion $p^{'} = \frac{x}{n}$ is calculated from the data.

$\frac{8}{40} = 0.2$

Part(b) Step 1: Given Information

The size of a simple random sample from a population, as well as the number of successes.

$x = 8, n = 40, 95 % level$

$∴ n - x = 40 - 8 = 32$ , here $x$ and $n - x$ are both $5$ or greater.

Part(b) step 2: Explanation

The confidence interval is $95 %$ , which means $α = 0.05$ .

It is discovered thatlocalid="1651326317744" $z_{a / 2} = z_{0.05 / 2} = 1.96$

The $p$ confidence interval is of the form

$p^{'} - z_{a α / 2} \sqrt{\frac{p^{'} (1 - p^{'})}{n}} to p^{'} + z_{a / 2} \sqrt{\frac{p^{'} (1 - p^{'})}{n}}$

i.e. localid="1651326715537" $0.2 - 1.96 \sqrt{\frac{0.2 (1 - 0.2)}{40}} to 0.2 + 1.96 \sqrt{\frac{0.2 (1 - 0.2)}{40}}$

i.e. localid="1651326727514" $0.2 - 0.124 to 0.2 + 0.124$

i.e.localid="1651326739386" $0.076 to 0.324$

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Short Answer

Step by step solution

Part(a) Step 1: Given Information

Part(a) Step 2: Explanation

Part(b) Step 1: Given Information

Part(b) step 2: Explanation

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