Chapter 5.2, Problem 35E

a.

To determine

Find the probability that exactly 18 students enrolled in college.

Answer to Problem 35E

The probability that exactly 18 students enrolled in college is 0.1166.

Explanation of Solution

Calculation:

It was found that 66% of the students graduated from high school enrolled in college.  A random sample of 30 graduates is considered.

Define the random variable X as the number of students graduated from high school enrolled in college.

Here, a random sample (n) of 30 graduates is taken. Each student is independent of the other. Also, there are two possible outcomes, the student graduated from high school enrolled in college or not (success or failure). It was found that 66% of the students graduated from high school enrolled in college. Thus, the probability of success (p) is 0.66. Hence, X follows binomial distribution.

The probability of obtaining x successes in n independent trails of a binomial experiment is,

$P\left(x\right)={}_n{C}_x{p}^x{\left(1-p\right)}^{n-x},x=0,1,2,\mathrm{...},n$

Where, p is the probability of success.

Substitute $n=30\text{ and }p=0.66$.

$P\left(x\right)={}_{30}{C}_x{\left(0.66\right)}^x{\left(1-0.66\right)}^{30-x}$

The probability that exactly 18 students enrolled in college is, $P\left(x=18\right)$.

Software procedure:

Step-by-step procedure to obtain the probability using MINITAB software is given below:

• Choose Calc > Probability Distributions > Binomial Distribution.
• Choose Probability.
• Enter Number of trials as 30 and Event probability as 0.66.
• In Input constant, enter 18.
• Click OK.

The output using the Minitab software is given below:

From the Minitab output, the probability value is approximately 0.1166.

Thus, the probability that exactly 18 students enrolled in college is 0.1166.

b.

To determine

Find the probability that more than 15 students enrolled in college.

Answer to Problem 35E

The probability that more than 15 students enrolled in college is 0.9486.

Explanation of Solution

Calculation:

The probability that more than 15 students enrolled in college is obtained as shown below:

$P\left(x>15\right)=1-P\left(x\le 15\right)$

Software procedure:

Step-by-step procedure to obtain the probability $P\left(x\le 15\right)$ using MINITAB software is given below:

• Choose Calc > Probability Distributions > Binomial Distribution.
• Choose Cumulative Probability.
• Enter Number of trials as 30 and Event probability as 0.66.
• In Input constant, enter 15.
• Click OK.

The output using the Minitab software is given below:

From the Minitab output, the probability value is approximately 0.0514. Substituting the value, the required probability becomes,

$\begin{array}{c}P\left(x>15\right)=1-P\left(x\le 15\right)\\ =1-0.0514\\ \approx 0.9486\end{array}$

Thus, the probability that more than 15 students enrolled in college is 0.9486.

c.

To determine

Find the probability that fewer than 12 students enrolled in college.

Answer to Problem 35E

The probability that fewer than 12 students enrolled in college is 0.0010.

Explanation of Solution

Calculation:

The probability that fewer than 12 students enrolled in college is obtained as shown below:

$P\left(x<12\right)=P\left(x\le 11\right)$

Software procedure:

Step-by-step procedure to obtain the probability $P\left(x\le 11\right)$ using MINITAB software is given below:

• Choose Calc > Probability Distributions > Binomial Distribution.
• Choose Cumulative Probability.
• Enter Number of trials as 30 and Event probability as 0.66.
• In Input constant, enter 11.
• Click OK.

The output using the Minitab software is given below:

From the Minitab output, the probability value is approximately 0.0010.

Thus, the probability that fewer than 12 students enrolled in college is 0.0010.

d.

To determine

Check whether it is unusual if more than 25 of them enrolled in college.

Answer to Problem 35E

Yes, it is unusual if more than 25 of them enrolled in college.

Explanation of Solution

Calculation:

Unusual:

If the probability of an event is less than 0.05 then the event is called unusual.

The probability that more than 25 of them enrolled in college is obtained as shown below:

 $P\left(x>25\right)=1-P\left(x\le 25\right)$

Software procedure:

Step-by-step procedure to obtain the probability $P\left(x\le 25\right)$ using MINITAB software is given below:

• Choose Calc > Probability Distributions > Binomial Distribution.
• Choose Cumulative Probability.
• Enter Number of trials as 30 and Event probability as 0.66.
• In Input constant, enter 25.
• Click OK.

The output using the Minitab software is given below:

From the Minitab output, the probability value is 0.9899. The required probability is,

$\begin{array}{c}P\left(x>25\right)=1-P\left(x\le 25\right)\\ =1-0.9899\\ =0.0101\end{array}$

Here, the probability of the event is less than 0.05. Thus, the event that more than 25 of them enrolled in college is unusual.

e.

To determine

Find the mean number of students who enrolled in college.

Answer to Problem 35E

The mean number of students who enrolled in college is 19.8.

Explanation of Solution

Calculation:

A binomial experiment with n independent trials and a probability of success p has mean,

${\mu }_X=np$

Substitute the values 30 for n and 0.66 for p,

The mean is,

$\begin{array}{c}{\mu }_X=np\\ =30\left(0.66\right)\\ =19.8\end{array}$

Thus, the mean number of students who enrolled in college is 19.8.

f.

To determine

Find the standard deviation of the number of students who enrolled in college in a sample of 30 students.

Answer to Problem 35E

The standard deviation of the number of students who enrolled in college in a sample of 30 students is 2.595.

Explanation of Solution

Calculation:

A binomial experiment with n independent trials and a probability of success p has standard deviation,

${\sigma }_X=\sqrt{np\left(1-p\right)}$

Substitute the values 30 for n and 0.66 for p,

$\begin{array}{c}{\sigma }_X=\sqrt{np\left(1-p\right)}\\ =\sqrt{30\left(0.66\right)\left(1-0.66\right)}\\ =\sqrt{6.732}\\ \approx 2.595\end{array}$

Thus, the standard deviation of the number of students who enrolled in college in a sample of 30 students is 2.595.