Chapter 5.3, Problem 78E

(a)

To determine

To find: The mean and standard deviation of the random variable X.

Answer to Problem 78E

Solution: The mean and standard deviation of the binomial random variable X is 996 and 13.012 respectively.

Explanation of Solution

Calculation: In binomial distribution, the mean can be calculated by this formula,

${\mu }_x=np$

In binomial distribution, standard deviation can be calculated by this formula,

${\sigma }_x=\sqrt{npq}$

Where, n is the number of trials and p is the probability of success in a trial. For $x=0,1,2,\mathrm{3...1200}$, the mean and standard deviation can be calculated by substituting values of n and p in the above formulas. The mean can be calculated as:

$\begin{array}{c}{\mu }_X=np\\ =1200\times 0.83\\ =996\end{array}$

The standard deviation is calculated as:

$\begin{array}{c}{\sigma }_X=\sqrt{np\left(1-p\right)}\\ =\sqrt{1200\left(0.83\right)\left(1-0.83\right)}\\ =13.012\end{array}$

Hence, the average value and standard deviation are 996 and 13.012 respectively.

(b)

To determine

To find: The probability.

Answer to Problem 78E

Solution: The probability is $\underset{\_}{P\left(X\ge 800\right)=1}$.

Explanation of Solution

Calculation: The binomial random variable approximately follows the normal distribution with mean $\mu =np$ and standard deviation $\sigma =\sqrt{np\left(1-p\right)}$ if the condition $np\left(1-p\right)\ge 10$ holds true, where n is the number of trials and p is the probability of success in a trial.

$\begin{array}{c}np\left(1-p\right)=1200\times 0.83\times \left(1-0.83\right)\\ =169.32\\ >10\end{array}$

Therefore, the random variable X can be approximated to normal distribution because the given condition is fulfilled.

The parameters of the normal distribution are calculated as follows:

$\begin{array}{c}\mu =np\\ =1200\times 0.83\\ =996\end{array}$

And:

$\begin{array}{c}\sigma =\sqrt{np\left(1-p\right)}\\ =\sqrt{1200\left(0.83\right)\left(1-0.83\right)}\\ =13.012\end{array}$

The probability that the value taken by X is greater than or equal to 800 is the area right to the point 800 under the normal curve. To obtain the area lying to the left of a point under the normal curve, first, convert the value of the variable into Z-score and then obtain the area lying to the right by subtracting the area left to the Z-score from 1. The Z-score for the value $X=800$ is calculated as:

$\begin{array}{c}Z=\frac{X-\mu }{\sigma }\\ =\frac{800-996}{13.012}\\ =-15.06\end{array}$

The area left to the particular Z-score can be obtained using Excel by using the command $\text{'}=\text{NORMSDIST}\left(\right)\text{'}$. Specify the value of z as $-15.06$.

The area left to the Z-score, $-15.06$, is obtained as 0.0000 from Excel.

So, the required probability can be calculated as:

$1-0.0000=1$

Hence, the probability is 1.

(c)

To determine

To find: The probability.

Answer to Problem 78E

Solution: $\underset{\_}{P\left(X>1000\right)=0.3676}$.

Explanation of Solution

Calculation: For the probability that more than 1000 candidates will accept the admission, Excel has to be used. The following procedure is followed in Excel:

Step 1: Open the Excel spreadsheet.

Step 2: Type the command $\text{'}=\text{BINOMDIST}\left(\right)\text{'}$ in one of the cells and specify the value of $\text{number}=1000,\text{trials = 1200, probability = 0}\text{.83,}\text{and cumulative as 'TRUE'}$. Press enter to obtain the probability that the value of the random variable X is less than or equal to 1000.

$P\left(X\le 1000\right)=0.6324$.

Therefore, the required probability is calculated as:

$\begin{array}{c}P\left(X>1000\right)=1-P\left(X\le 1000\right)\\ =1-0.6324\\ =0.3676\end{array}$

Hence, the probability is 0.3676.

(d)

To determine

To find: The probability.

Answer to Problem 78E

Solution: $\underset{\_}{P\left(X>1000\right)=0.0001}$.

Explanation of Solution

Calculation: The probability that more than 1000 candidates will accept the admission, Excel has to be used. The following procedure is followed in Excel:

Step 1: Open the Excel spreadsheet.

Step 2: Type the command $\text{'}=\text{BINOMDIST}\left(\right)\text{'}$ in one of the cell and specify the value of $\text{number}=1000,\text{trials = 1150, probability = 0}\text{.83,}\text{and cumulative as 'TRUE'}$. Press enter to obtain the probability that the value of the random variable X is less than or equal to 1000.

$P\left(X\le 1000\right)=0.9999$

Therefore, the required probability is calculated as:

$\begin{array}{c}P\left(X>1000\right)=1-P\left(X\le 1000\right)\\ =1-0.9999\\ =0.0001\end{array}$

Hence, the probability is 0.0001.