Chapter 12, Problem 12.35E

(a)

To determine

To find out what are n and p , X have a binomial distribution.

Answer to Problem 12.35E

  $n=6\text{ and }p=0.75$ .

Explanation of Solution

In the question, it is given that the West African country of Guinea has the highest rate of malaria in the world with $75\%$ of its population being infected. And let X be the count of patients who are infected with malaria among the $6$ patients the doctor saw. Thus, we know that X follows binomial distribution then we have sample size of $6$ and the population proportion will be the probability of population being infected, $75\%$ . Thus, we have,

  $X\sim Bin(6,0.75)$ .

(b)

To determine

To find out what are the possible values that X can take.

Answer to Problem 12.35E

  $S=\{0,1,2,3,4,5,6\}$ .

Explanation of Solution

In the question, it is given that the West African country of Guinea has the highest rate of malaria in the world with $75\%$ of its population being infected. And let X be the count of patients who are infected with malaria among the $6$ patients the doctor saw. Thus, we know that X follows binomial distribution with,

  $X\sim Bin(6,0.75)$

Thus, the possible values for X will be as:

  $S=\{0,1,2,3,4,5,6\}$ as the sample size is six.

(c)

To determine

To find the probability of each value of X and draw a probability histogram of X .

Answer to Problem 12.35E

X	Exact probability
0	0.000244
1	0.004395
2	0.032959
3	0.131836
4	0.296631
5	0.355957
6	0.177979

Explanation of Solution

In the question, it is given that the West African country of Guinea has the highest rate of malaria in the world with $75\%$ of its population being infected. And let X be the count of patients who are infected with malaria among the $6$ patients the doctor saw. Thus, we know that X follows binomial distribution with,

  $X\sim Bin(6,0.75)$

Thus, the probability of each value of X can be calculated by using the binomial excel function:

  $=\text{BINOM}\text{.DIST(numbers, trials, prob, cumulative)}$

In the cumulative, the FALSE gives the exact value and TRUE gives the less than value.

Thus, the calculation will be as:

X	Exact probability
0	=BINOM.DIST(AJ116,6,0.75,FALSE)
1	=BINOM.DIST(AJ117,6,0.75,FALSE)
2	=BINOM.DIST(AJ118,6,0.75,FALSE)
3	=BINOM.DIST(AJ119,6,0.75,FALSE)
4	=BINOM.DIST(AJ120,6,0.75,FALSE)
5	=BINOM.DIST(AJ121,6,0.75,FALSE)
6	=BINOM.DIST(AJ122,6,0.75,FALSE)

The result will be as:

X	Exact probability
0	0.000244
1	0.004395
2	0.032959
3	0.131836
4	0.296631
5	0.355957
6	0.177979

Thus, the probability histogram of X will be as:

(d)

To determine

To find out what are the mean and standard deviation of this distribution and mark the location of the mean on your histogram.

Answer to Problem 12.35E

The mean is $4.5$ and the standard deviation is $1.061$ .

Explanation of Solution

In the question, it is given that the West African country of Guinea has the highest rate of malaria in the world with $75\%$ of its population being infected. And let X be the count of patients who are infected with malaria among the $6$ patients the doctor saw. Thus, we know that X follows binomial distribution with,

  $X\sim Bin(6,0.75)$

Thus, the mean and standard deviation of this distribution will be calculated as:

  $\begin{array}{l}\text{Mean}=np\\ =6\times 0.75\\ =4.5\\ \text{St}\text{. Dev}\text{.}=\sqrt{npq}\\ =\sqrt{6\times 0.75\times (1-0.75)}\\ =1.061\end{array}$

Thus, the mean in the histogram is shown as: