Chapter 6, Problem 8CRE

a.

To determine

To explain: The mean and standard deviation of number of heads.

Answer to Problem 8CRE

The mean and standard deviation are 5000 and 50 respectively.

Explanation of Solution

Given:

Number of trials (n) = 10000

Probability of success (p) = 0.5

Formula used:

The formula to compute the mean and standard deviation using binomial distribution is:

  $\begin{array}{l}\mu =n\times p\\ \sigma =\sqrt{n\times p\times \left(1-p\right)}\end{array}$

Calculation:

The mean and standard deviation are:

  $\begin{array}{l}\mu =n\times p=10000\left(0.5\right)=5000\\ \sigma =\sqrt{n\times p\times \left(1-p\right)}=\sqrt{10000\left(0.5\right)\times \left(1-0.5\right)}=50\end{array}$

Thus, the mean and standard deviation are 5000 and 50.

b.

To determine

To explain: The reason of using normal approximation.

Answer to Problem 8CRE

Explanation of Solution

Calculation:

Here,

  $\begin{array}{l}np=10000\left(0.5\right)=5000\ge 5\\ n\left(1-p\right)=10000\left(1-0.5\right)=5000\ge 5\end{array}$

Thus, normal approximation could be used here.

c.

To determine

To explain: If there is reason to believe that Kerrich coin was not balanced by computing probability.

Answer to Problem 8CRE

No.

Explanation of Solution

Calculation:

The probability is calculated as:

  $\begin{array}{c}P\left(X\le 4933\text{ or }X\ge 5067\right)=P\left(\frac{x-\mu }{\sigma }\le \frac{4933-5000}{500}\text{or }\frac{x-\mu }{\sigma }\ge \frac{5063-5000}{500}\right)\\ =P\left(Z\le -1.34\text{ or }Z\ge 1.34\right)\\ =2\times P\left(Z<-1.34\right)\\ =0.1802\end{array}$

Since, the probability is above 5% thus there is no reason to believe that Kerrich’s coin was not balanced by computing probability.