Chapter 9.1, Problem 9.20E

a)

To determine

To find the sample proportion.

Answer to Problem 9.20E

Sample proportion is 0.3534

Explanation of Solution

Given:

n = sample size = 928

  $x$ = Number of females = 328

Formula:

  $\widehat{p}=\frac{x}{n}$

First need to understand about sample and population.

Population: It is the set of all the possible individuals possessing the characteristic of interest in a study.

Sample: it is the subset of population on which, the actual study performed and from which, the necessary measurements are taken.

Sample proportion is the ratio of number of successes divided by the sample size.

Therefore,

  $\widehat{p}=\frac{328}{928}=0.3534$

b)

To determine

To find the 95% confidence interval for p

Answer to Problem 9.20E

The 95% confidence interval is, (3226,3842)

Explanation of Solution

Given:

C=confidence level = 0.95

n = sample size = 928

  $x$ = Number of females = 328

Formula:

Confidence interval for population proportion is,

  $CI=\widehat{p}\mp E$

Sample proportion:

  $\widehat{p}=0.3534$

Margin of error:

  $E=Zc\times \sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}$

Calculation:

Z critical value can be found either by using standard normal table or excel formula.

Therefore, level of significance = a = 1-0.95 = 0.05

So, Z_c =Z_a/2 = Z_0.05/2

Using excel function, =ABS(NORMSINV(0.05/2))

Zc = 1.96

Therefore, margin of error is,

  $E=1.96\times \sqrt{\frac{0.3534(1-0.3534)}{928}}=0.0308$

95% confidence interval for population proportion,

  $\begin{array}{l}\widehat{p}-E=0.3534-0.0308=0.3226\\ \widehat{p}+E=0.3534+0.0308=0.3842\end{array}$

We are 95% confident that the true proportion falls between 0.3226 and 0.3842

c)

To determine

To explain factors necessary to find margin of error.

Explanation of Solution

Given:

E = Margin of error = 0.0308

Formula:

Margin of error:

  $E=Zc\times \sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}$

Calculation:

There are three factors necessary to determine the margin of error instead of two factors. The confidence level, sample proportion and sample size.