Chapter 7.3, Problem 13P

(a)

To determine

Find the point estimate for p.

Answer to Problem 13P

The point estimate for p is 0.3100.

Explanation of Solution

Calculation:

Point estimate:

The point estimate for p is,

$\widehat{p}=\frac{r}{n}$

In the formula n is the number of trails, and r is the number of success.

Let p represent the proportion of all permanent dwellings on the entire NavajoReservation that are traditional hogans.

A random sample of 5,222 permanentdwellingsis considered in which1,619 were traditional Navajo hogans. Thatis $n=5,222$, $r=1,619$.

Substitute 1,619 as r, and 5,222 as n in the point estimate formula.

$\begin{array}{c}\widehat{p}=\frac{1,619}{5,222}\\ =0.3100\end{array}$

Hence, the point estimate for p is 0.3100.

(b)

To determine

Find the 99% confidence interval for p.

Interpret the confidence interval.

Answer to Problem 13P

The 99% confidence interval for p is $0.2935<p<0.3265$.

Explanation of Solution

Calculation:

Confidence interval:

The confidence interval formula is,

$\widehat{p}-E<p<\widehat{p}+E$

In the formula $E=z_c\sqrt{\frac{\widehat{p}\widehat{q}}{n}}$, $\widehat{p}$ is probability of success, $\widehat{q}$ is probability of failure, n is the sample size, c is confidence level, and $z_c$ is critical value for confidence level c based on the standard normal distribution.

The confidence level is 99%.

Critical value:

Use the Appendix II: Tables, Table 5 (b): Confidence interval, Critical Values $z_c$.

• In the level of confidence c column, locate the value of 0.99, or 99%.
• The correspondingCritical value $z_c$ is2.58.

The error is,

$\begin{array}{c}E=2.58\sqrt{\frac{\left(0.3100\right)\left(1-0.3100\right)}{5,222}}\\ =2.58\times 0.0064\\ =0.0165\end{array}$

Confidence interval:

Substitute $\widehat{p}$ as 0.3100, E as 0.0165 in the confidence interval formula

$\begin{array}{c}0.3100-0.0165<p<0.3100+0.0165\\ 0.2935<p<0.3265\end{array}$

Hence, the 99% confidence interval for p is $0.2935<p<0.3265$.

The confidence interval can be interpreted as; there is 99% confidence that the population proportion of all permanent dwellings on the entire NavajoReservation that are traditional hogans lies within the interval 0.2935 to 0.3265.

(c)

To determine

Identify whether the conditions $n\widehat{p}>5$ and $n\widehat{q}>5$ are satisfied or not.

Describe why satisfying conditions would be an important consideration.

Explanation of Solution

Checking conditions:

$\begin{array}{c}n\widehat{p}=5,222\left(0.3100\right)\\ =1,619\\ >5\end{array}$

$\begin{array}{c}n\widehat{q}=n\left(1-\widehat{p}\right)\\ =5,222\left(1-0.3100\right)\\ =5,222\left(0.6900\right)\\ =3,603\\ >5\end{array}$

It can be observed that two of the conditions $n\widehat{p}>5$, $n\widehat{q}>5$ are satisfied.

Conditions for normal approximation to $\widehat{p}$ distribution:

For a binomial experiment with n number of trails, r number of success, probability of success for each trail $\widehat{p}$ and probability of failure $\widehat{q}=1-\widehat{p}$, the $\widehat{p}=\frac{r}{n}$ is approximated to a normal distribution if,

• $n\widehat{p}>5$
• $n\widehat{q}>5$

The probability of success for each trail $\widehat{p}$ is normally distributed with $\mu =p$ and $\sigma =\sqrt{\frac{pq}{n}}$ when the conditions are satisfied. This shows that, it is necessary that the conditions $n\widehat{p}>5$ and $n\widehat{q}>5$ must be satisfied to use approximation.