Chapter 6.4, Problem 6.38E

Let x be a binomial random variable with $n=15$ and $p=.5$ .

a. Is the normal approximation appropriate?

b. Find $P\left(x\ge 6\right)$ using the normal approximation.

c. Find $P\left(x>6\right)$ using the normal approximation.

d. Find the exact probabilities for parts b and c, and compare these with your approximations.

To determine

To find: Whether the normal approximation will be appropriate.

Answer to Problem 6.38E

The normal approximation will be appropriate.

Explanation of Solution

Given information: The values of $n$ is $15$ and value of $p$ is $0.5$ .

Calculation:

The value of $np$ is,

  $\begin{array}{c}np=15\times 0.5\\ =7.5\end{array}$

The value of $nq$ is,

  $\begin{array}{c}nq=15\times \left(1-0.5\right)\\ =7.5\end{array}$

Since, the value $np$ and $nq$ are greater than five.

Thus, the normal approximation will be appropriate.

To determine

To find: The value of $P\left(x\ge 6\right)$ .

Answer to Problem 6.38E

The value of $P\left(x\ge 6\right)$ is $0.8485$ .

Explanation of Solution

Given information: The values of $n$ is $15$ and value of $p$ is $0.5$ .

Calculation:

The mean is,

  $\begin{array}{c}\mu =np\\ =15\times 0.5\\ =7.5\end{array}$

The standard deviation is,

  $\begin{array}{c}\sigma =\sqrt{npq}\\ =\sqrt{15\times 0.5\times 0.5}\\ =1.94\end{array}$

The $z$ score is,

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ =\frac{5.5-7.5}{1.94}\\ =-1.03\end{array}$

The probability is,

  $\begin{array}{c}P\left(x\ge 6\right)=P\left(z\ge -1.03\right)\\ =1-P\left(z<-1.03\right)\\ =1-0.1515\\ =0.8485\end{array}$

Thus, the value of $P\left(x\ge 6\right)$ is $0.8485$ .

To determine

To find: The value of $P\left(x>6\right)$ .

Answer to Problem 6.38E

The value of $P\left(x>6\right)$ is $0.6985$ .

Explanation of Solution

Given information: The values of $n$ is $15$ and value of $p$ is $0.5$ .

Calculation:

The $z$ score is,

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ =\frac{6.5-7.5}{1.94}\\ =-0.52\end{array}$

The probability is,

  $\begin{array}{c}P\left(x>6\right)=P\left(z>-0.52\right)\\ =1-P\left(z\le -0.52\right)\\ =1-0.3015\\ =0.6985\end{array}$

Thus, the value of $P\left(x>6\right)$ is $0.6985$ .

To determine

To find: The exact value of $P\left(x\ge 6\right)$ and $P\left(x>6\right)$ .

Answer to Problem 6.38E

The exact value of $P\left(x\ge 6\right)$ and $P\left(x>6\right)$ is $0.849$ and $0.696$ .

Explanation of Solution

Given information: The values of $n$ is $15$ and value of $p$ is $0.5$ .

Calculation:

The probability using the table is,

  $\begin{array}{c}P\left(x\ge 6\right)=1-P\left(x<6\right)\\ =1-0.151\\ =0.849\end{array}$

The probability using the table is,

  $\begin{array}{c}P\left(x>6\right)=1-P\left(x\le 6\right)\\ =1-0.304\\ =0.696\end{array}$

Thus, the exact value of $P\left(x\ge 6\right)$ and $P\left(x>6\right)$ is $0.849$ and $0.696$ .