According to The World Bank, only 9% of the population of Uganda had access to electricity as of 2009. Suppose we randomly sample 194 people in Uganda. Let X = the number of people who have access to electricity. a. What is the probability distribution for X?
b. Using the formulas, calculate the mean and standard deviation of X.
c. Use your calculator to find the probability that 15 people in the sample have access to electricity.
d. Find the probability that at most ten people in the sample have access to electricity.
e. Find the probability that more than 25 people in the sample have access to electricity.

Step-by-step explanation:

Hello!

The variable of interest is:

X: Number of people in Uganda who have access to electricity out of 194.

a.

This variable is discrete, to check if it has a binomial distribution I'll check if it follows the binomial criteria:

1. The number of observation of the trial is fixed n= 194

2. Each observation in the trial is independent, this means that none of the trials will affect the probability of the next trial.

3. The trial has only two possible outcomes, "success": the person has access to electricity and "failure": the person does not have access to electricity.

4. The probability of success in the same from one trial to another. In this trial, the probability of success is the proportion of the Ugandan population that has access to electricity p=0.09

So X≈ Bi (n;ρ)

Where n represents the sample (n=194) and ρ is the probability of success (ρ=0.09)

b.

Under the binomial distribution you can calculate the mean as:

E(X)= n*p= 194*0.09= 17.46

And the variance is:

V(X)= n*p*(1-p)= 194*0.09*0.91= 15.8886

The Standard deviation is the square root of the variance:

√V(X)= 3.986≅ 3.99

c.

To reach the probability values I usually use the Binomial tables of accumulated probabilities, but to calculate a punctual probability for a certain value you can also use the formula:

P(X=15)= 0.0874

Using the table pf accumulative probabilities, to reach the probability of finding exactly 15 people with access to electricity, you have to subtract the accumulated probability until 14 to the accumulated probability until 15:

P(X=15)= P(X≤15) - P(X≤14)= 0.3209 - 0.2335= 0.0874

d.

When you are looking for the probability of the value of X being "at most" 10, this means that you are looking for the probability of X being equal or less to 10:

P(X≤10)= 0.0332

You can look for this value of probability directly in the table.

e.

P(X>25)

Using the table you have to first do the following conversion, think that if you have the event "X being greater than 25" the complementary event would be that "X is equal or less than 25" then you can say that that the probabilities are:

P(X>25)= 1 - P(X≤25)= 1 - 0.9733= 0.0267

I hope it helps!