Chapter 6, Problem R6.6RE

(a)

To determine

Interpretation of expected value of X .

Answer to Problem R6.6RE

Expected value of X ,

  ${\mu }_X=1.64\text{candies}$

Explanation of Solution

Given information:

Number of trials, $n=8$

Probability of success, $p=20.5\%=0.205$

The number of successes among a fixed number of independent trials follows a binomial distribution.

Then

The expected mean (or value) of a binomial distribution is the product of sample size ( n ) and probability ( p ).

  $\mu =np=8\times 0.205=1.64$

Thus,

On an average, 1.64 candies of the 8 randomly selected candies are orange.

(b)

To determine

Interpretation of standard deviation of X .

Answer to Problem R6.6RE

Standard deviation of X ,

  ${\sigma }_X\approx 1.418\text{candies}$

Explanation of Solution

Given information:

Number of trials, $n=8$

Probability of success, $p=20.5\%=0.205$

The standard deviation of a binomial distribution is the square root of the product of the sample size ( n ) and the probabilities ( p ) and ( q ).

  $\sigma =\sqrt{npq}=\sqrt{np\left(1-p\right)}=\sqrt{8\left(0.205\right)\left(1-0.205\right)}=\sqrt{1.3038}\approx 1.1418$

Thus,

On an average, the number of orange candies among 8 randomly selected candies varies by approx. 1.1418 candies from the mean of 1.64 candies.