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Q. 102

Expert-verified
Found in: Page 295

### Introductory Statistics

Book edition OER 2018
Author(s) Barbara Illowsky, Susan Dean
Pages 902 pages
ISBN 9781938168208

a. The probability distribution of X is $0.09$

b. The mean is $13.5$and standard deviation is $3.5$

c. The probability that $15$ people in the sample have access to electricity is $0.0988$

d. The probability that at most ten people in the sample have access to electricity is $0.1987$

e. The probability that more than $25$ people in the sample have access to electricity is $0.9991$

See the step by step solution

## Step 1: Content Introduction

The binomial distribution determines the probability of looking at a specific quantity of a hit results in a specific quantity of trials

## Step 2: Explanation (part a)

We are given,

$9%$of the population of Uganda had access to electricity as of 2009 and we randomly chose a sample of 150 people in Uganda. So. here $n$ is the sample size and $p$ is the probability.

Therefore,

The probability distribution of X is $p=0.09$

## Step 2: Explanation (part b)

We are given: $n=150,p=0.09$

The mean is:

$\mu =np$, where

$n=numberofsamples\phantom{\rule{0ex}{0ex}}p=probabilityofsamples$

Therefore, the mean is

role="math" localid="1649250625500" $\mu =np\phantom{\rule{0ex}{0ex}}\mu =150×0.09\phantom{\rule{0ex}{0ex}}\mu =13.5$

Standard deviation is:

$\sigma =\sqrt{np\left(1-p\right)}\phantom{\rule{0ex}{0ex}}\sigma =\sqrt{150×0.09\left(1-0.09\right)}\phantom{\rule{0ex}{0ex}}\sigma =3.5$

## Step 4: Explanation (part c)

Using binompdf:

the probability that $15$ people in the sample have access to electricity is:

role="math" localid="1649250800761" $binompdf\left(150,0.09,15\right)=\frac{150!}{15!\left(150-15\right)!}0.{09}^{15}{\left(1-0.09\right)}^{50-15}\phantom{\rule{0ex}{0ex}}binompdf\left(150,0.09,15\right)=\frac{150!}{15!\left(150-15\right)!}0.{09}^{15}{\left(1-0.09\right)}^{35}\phantom{\rule{0ex}{0ex}}binompdf\left(150,0.09,15\right)=0.0988$

## Step 5: Explanation (part d)

The probability that at most ten people in the sample have access to electricity is:

$binompdf\left(150,0.09,10\right)=\sum _{k=0}^{10}\frac{150!}{!\left(k150-k\right)!}0.{09}^{k}{\left(1-0.09\right)}^{50-k}\phantom{\rule{0ex}{0ex}}binompdf\left(150,0.09,10\right)=0.1897$

## Step 6: Explanation (part e)

The probability that more than 25 people in the sample have access to electricity is:

$binompdf\left(150,0.09,25\right)=\sum _{k=0}^{25}\frac{150!}{!\left(k150-k\right)!}0.{09}^{k}{\left(1-0.09\right)}^{50-k}\phantom{\rule{0ex}{0ex}}binompdf\left(150,0.09,25\right)=0.9991$