Question

The purpose of this Question is to investigate some of the properties of the binomial probability distribution using Microsoft Excel. a. Suppose that for a binomial experiment with n = 15 trials, the probability of success is p = 0.10, and x is the number of successes. Obtain the probability distribution of x. Make a histogram for the probability distribution and comment on its shape. Find the mean and standard deviation of x. b. Suppose that for a binomial experiment with n = 15 trials, the probability of success is p = 0.50, and x is the number of successes. Obtain the probability distribution of x. Make a histogram for the probability distribution and comment on its shape. Find the mean and standard deviation of x. c. Which of the distributions in parts a–b had the greatest variability in the number of successes? Why does this make sense?

Answer 1

Result:

The purpose of this Question is to investigate some of the properties of the binomial probability distribution using Microsoft Excel.

a. Suppose that for a binomial experiment with n = 15 trials, the probability of success is p = 0.10, and x is the number of successes. Obtain the probability distribution of x. Make a histogram for the probability distribution and comment on its shape. Find the mean and standard deviation of x.

Expectation = np = 1.5

Variance = np(1 - p) = 1.35

Standard deviation = 1.1619

Binomial Probabilities
Data
Sample size	15
Probability of an event of interest	0.10
Statistics
Mean	1.5
Variance	1.3500
Standard deviation	1.1619
Binomial Probabilities Table
	X	P(X)
	0	0.2059
	1	0.3432
	2	0.2669
	3	0.1285
	4	0.0428
	5	0.0105
	6	0.0019
	7	0.0003
	8	0.0000
	9	0.0000
	10	0.0000
	11	0.0000
	12	0.0000
	13	0.0000
	14	0.0000
	15	0.0000

The shape of the distribution is right skewed.

b. Suppose that for a binomial experiment with n = 15 trials, the probability of success is p = 0.50, and x is the number of successes. Obtain the probability distribution of x. Make a histogram for the probability distribution and comment on its shape. Find the mean and standard deviation of x.

Binomial Probabilities
Data
Sample size	15
Probability of an event of interest	0.5
Statistics
Mean	7.5
Variance	3.7500
Standard deviation	1.9365
Binomial Probabilities Table
	X	P(X)
	0	0.0000
	1	0.0005
	2	0.0032
	3	0.0139
	4	0.0417
	5	0.0916
	6	0.1527
	7	0.1964
	8	0.1964
	9	0.1527
	10	0.0916
	11	0.0417
	12	0.0139
	13	0.0032
	14	0.0005
	15	0.0000

The shape of the distribution is bell shaped, symmetric.

c. Which of the distributions in parts a–b had the greatest variability in the number of successes? Why does this make sense?

The distribution with in part b ( n=15, p=0.50)had the greatest variability in the number of successes. This is because , for the same n > 1, as P tends to zero, variance, P(1 – P) will also tend to 0. Variance is maximum when p is 0.5.