91Ó°ÊÓ

The probability of success is $x=10$, the sample size of a basic random sampling from a population is $40$, and the probability of failure is $90\%$.

The sample proportion is the percentage of the total number of members sampled who have a specific attribute and a sample size of $n$.

In the formula, replace $x$with $10$and $n$with $40$.

$p^{âˆ\S }=10/40=0.25$

The probability of success is $x=10$, the sample size of a basic random sampling from a population is $40$, and the probability of failure is $90\%$.

The confidence interval for a population percentage, $p$, is calculated using the one-proportion $z$-interval procedure.

Assumptions:

- A simple random sample should be used.

- Both the number of successes $x$and failures $n-x$should be $5$or larger.

The number of successes, in this case, is $x=10$, which is larger than $5$.

The number of failures is,

$$\begin{gathered} n-x=40-10 \\ =30 \end{gathered}$$

The number of failures $n-x$exceeds $5$.

The probability of success is $x=10$, the sample size of a basic random sampling from a population is $40$, and the probability of failure is $90\%$.

The One-Proportion $z$-interval Procedure is applicable, as shown in section (b).

The $90\%$confidence interval is as follows:

The formula for calculating the confidence interval for $p$is as follows:

$p^{âˆ\S }\text{is}0.25$ based on component (a).

In the formula, replace $\mathrm{Î\pm }$with $0.10$and $n$with $40$.

The value $z_{0.05}=1.645$.

$\hat{p}Â\pm z_{\frac{a}{2}}·\sqrt{\hat{p}(1-\hat{p})/n}=0.25Â\pm 1.645·\sqrt{0.25(0.75)/40}$

$=0.25Â\pm 1.645·\sqrt{0.1875/40}$

$=0.25Â\pm 1.645·\sqrt{0.0046875}$

$=0.25Â\pm 1.645·(0.0685)$

$=0.25Â\pm 0.1127$

$=(0.25-0.1127,0.25+0.1127)$

$=(0.1373,0.3627)$

$≈(0.137,0.363)$

Thus, the $90\%$confidence interval is $(0.137,0.363)$.

The probability of success is $x=10$, the sample size of a basic random sampling from a population is $40$, and the probability of failure is $90\%$.

The One-Proportion $z$-interval Procedure is applicable, as shown in section (b).

The confidence interval is $90\%$, $\mathrm{Î\pm }=0.10$, and the sample percentage is $0.25$from component (a).

In the formula, replace $\mathrm{Î\pm }$with $0.10$, $p^{âˆ\S }$with $0.25$, and $n$with $40$.

$E=z_{\frac{0.19}{2}}·\sqrt{\frac{0.25(1-0.25)}{40}}$

$=z_{0.05}·\sqrt{\frac{0.25(0.75)}{40}}$

$=z_{0.05}·\sqrt{\frac{0.1875}{40}}$

$=z_{0.05}·\sqrt{0.0046875}$

From the bottom of Table IV: Values of$t_{\mathrm{Î\pm }}$

The value $z_{0.05}=1.645$.

$$\begin{gathered} E=1.645·0.0685 \\ =0.1127 \end{gathered}$$