Question

The literacy rate for a nation measures the proportion of people age 15 and over who can read and write. The literacy rate for women in Afghanistan is 12%. Let X = the number of Afghani women you ask until one says that she is literate.

a. What is the probability distribution of X?

b. What is the probability that you ask five women before one says she is literate?

c. What is the probability that you must ask ten women?

d. Find the (i) mean and (ii) standard deviation of X.

Short answer

a. X follows geometric distribution with the parameters $X~G(0.12)$

b. The probability that you ask five women before one says she is literate $0.4723$

c. The probability that you must ask ten women $0.7125$

d.$$\begin{gathered} mean\mu =8.333 \\ s\tan darddeviation\delta =7.82 \end{gathered}$$

Step-by-step solution

Content Introduction

We are given,

The literacy rate for women in Afghanistan is $12\%$

X = the number of Afghani women

Explanation (Part a)

Let X the number of women ask until one says she is literature. The sample size X takes on the values $0,1,2,3,......$with probability $p=0.12$

Therefore, X follows geometric distribution with the parameters,$X~G(0.12)$

Explanation (Part b)

Let X the number of women ask until one says she is literature. The sample size X takes on the values $0,1,2,3,......$with probability $p=0.12$

Therefore, X follows geometric distribution with the parameters, $X~G(0.12)$

To find the probability that $X=5$step in T1-83+ calculator is as follow:

Press $1$and then press $2^{nd}$DISTR

Arrow down to geometpdf. Press ENTER

Enter $(0.12,5)$

The result shows$P(X=5)=0.4723$

Explanation (Part c)

Let X the number of women ask until one says she is literature. The sample size X takes on the values $0,1,2,3,......$with probability $p=0.12$

Therefore, X follows geometric distribution with the parameters $X~G(0.12)$

To find the probability that $X=10$step in T1-83+ calculator is as follow:

Press $1$and then press $2^{nd}$DISTR

Arrow down to geometpdf. Press ENTER

Enter $(0.12,10)$

The result shows$P(X=10)=0.7215$

Explanation (Part d)

Let X the number of women ask until one says she is literature. The sample size X takes on the values $0,1,2,3,......$with probability $p=0.12$

Therefore, X follows geometric distribution with the parameters $X~G(0.12)$

$$\begin{gathered} Mean\left(\mu \right)=\frac{1}{p} \\ s\tan darddeviation\left(\delta \right)=\sqrt{\frac{1-p}{p^2}} \end{gathered}$$

Substitute the value

$$\begin{gathered} mean\left(\mu \right)=\frac{1}{0.12} \\ mean\left(\mu \right)=8.333 \\ S\tan darddeviation\left(\delta \right)=\sqrt{\frac{1-0.12}{{(0.12)}^2}} \\ S\tan darddeviation\left(\delta \right)=7.82 \end{gathered}$$