Chapter 5.2, Problem 36E

a.

To determine

Find the probability that exactly 5 babies weigh more than 20 pounds.

Answer to Problem 36E

The probability that exactly 5 babies weigh more than 20 pounds is 0.1802.

Explanation of Solution

Calculation:

It was found that 25% of the babies weigh more than 20 pounds.  A random sample of 16 babies is considered.

Define the random variable X as the number of babies weigh more than 20 pounds. Here, a random sample (n) of 16 babies is taken. Each baby is independent of the other. Also, there are two possible outcomes, the baby weigh more than 20 pounds or not (success or failure). It was found that 25% of the babies weigh more than 20 pounds. Thus, the probability of success (p) is 0.25. 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 $n=16\text{ and }p=0.25$.

$P\left(x\right)={}_{16}{C}_x{\left(0.25\right)}^x{\left(1-0.25\right)}^{16-x}$

The probability that exactly 5 babies weigh more than 20 pounds is, $P\left(x=5\right)$.

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 16 and Event probability as 0.25.
• In Input constant, enter 5.
• Click OK.

The output using the Minitab software is given below:

From the Minitab output, the probability value is approximately 0.1802.

Thus, the probability that exactly 5 babies weigh more than 20 pounds is 0.1802.

b.

To determine

Find the probability that more than 6 babies weigh more than 20 pounds.

Answer to Problem 36E

The probability that more than 6 babies weigh more than 20 pounds is 0.0796.

Explanation of Solution

Calculation:

The probability that more than 6 babies weigh more than 20 pounds is obtained as shown below:

$P\left(x>6\right)=1-P\left(x\le 6\right)$

Software procedure:

Step-by-step procedure to obtain the probability $P\left(x\le 6\right)$ using MINITAB software is given below:

• Choose Calc > Probability Distributions > Binomial Distribution.
• Choose Cumulative Probability.
• Enter Number of trials as 16 and Event probability as 0.25.
• In Input constant, enter 6.
• Click OK.

The output using the Minitab software is given below:

From the Minitab output, the probability value is approximately 0.9204. Substituting the value, the required probability becomes,

$\begin{array}{c}P\left(x>6\right)=1-P\left(x\le 6\right)\\ =1-0.9204\\ \approx 0.0796\end{array}$

Thus, the probability that more than 6 babies weigh more than 20 pounds is 0.0796.

c.

To determine

Find the probability that fewer than 3 babies weigh more than 20 pounds.

Answer to Problem 36E

The probability that fewer than 3 babies weigh more than 20 pounds is 0.1971.

Explanation of Solution

Calculation:

The probability that fewer than 3 babies weigh more than 20 pounds is obtained as shown below:

$P\left(x<3\right)=P\left(x\le 2\right)$

Software procedure:

Step-by-step procedure to obtain the probability $P\left(x\le 2\right)$ using MINITAB software is given below:

• Choose Calc > Probability Distributions > Binomial Distribution.
• Choose Cumulative Probability.
• Enter Number of trials as 16 and Event probability as 0.25.
• In Input constant, enter 2.
• Click OK.

The output using the Minitab software is given below:

From the Minitab output, the probability value is approximately 0.1971.

Thus, the probability that fewer than 3 babies weigh more than 20 pounds is 0.1971.

d.

To determine

Check whether it is unusual if more than 8 of the babies weigh more than 20 pounds.

Answer to Problem 36E

Yes, it is unusual if more than 8 of the babies weigh more than 20 pounds.

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 8 of the babies weigh more than 20 pounds is obtained as shown below:

 $P\left(x>8\right)=1-P\left(x\le 8\right)$

Software procedure:

Step-by-step procedure to obtain the probability $P\left(x\le 8\right)$ using MINITAB software is given below:

• Choose Calc > Probability Distributions > Binomial Distribution.
• Choose Cumulative Probability.
• Enter Number of trials as 16 and Event probability as 0.25.
• In Input constant, enter 8.
• Click OK.

The output using the Minitab software is given below:

From the Minitab output, the probability value is 0.9925. The required probability is,

$\begin{array}{c}P\left(x>8\right)=1-P\left(x\le 8\right)\\ =1-0.9925\\ =0.0075\end{array}$

Here, the probability of the event is less than 0.05. Thus, the event that more than 8 of the babies weigh more than 20 pounds is unusual.

e.

To determine

Find the mean number of babies who weigh more than 20 pounds in a sample of 16 babies.

Answer to Problem 36E

The mean number of babies who weigh more than 20 pounds in a sample of 16 babies is 4.

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 16 for n and 0.25 for p,

The mean is,

$\begin{array}{c}{\mu }_X=np\\ =16\left(0.25\right)\\ =4\end{array}$

Thus, the mean number of babies who weigh more than 20 pounds in a sample of 16 babies is 4.

f.

To determine

Find the standard deviation of the number of babies who weigh more than 20 pounds in a sample of 16 babies.

Answer to Problem 36E

The standard deviation of the number of babies who weigh more than 20 pounds in a sample of 16 babies is 1.7321.

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 16 for n and 0.25 for p,

$\begin{array}{c}{\sigma }_X=\sqrt{np\left(1-p\right)}\\ =\sqrt{16\left(0.25\right)\left(1-0.25\right)}\\ =\sqrt{3}\\ \approx 1.7321\end{array}$

Thus, the standard deviation of the babies who weigh more than 20 pounds in a sample of 16 babies is 1.7321.