Question

1. Calculate (by hand) the mean and standard deviation for the binomial distribution with the probability of a success being 0.50 and n = 10. Either show work or explain how your answer was calculated. Use these formulas to do the hand calculations: Mean = np, Standard Deviation =  Comparison:
2. Calculate (by hand) the mean and standard deviation for the binomial distribution with the probability of a success being 0.10 and n = 10. Write a comparison of these statistics to those from question 5 in a short paragraph of several complete sentences. Use these formulas to do the hand calculations: Mean = np, Standard Deviation = Comparison:
3. Calculate (by hand) the mean and standard deviation for the binomial distribution with the probability of a success being 0.90 and n = 10. Write a comparison of these statistics to those from question 6 in a short paragraph of several complete sentences. Use these formulas to do the hand calculations: Mean = np, Standard Deviation =   
   
     

Answer 1

5.

Probability of success, p = 0.50

Sample size, n = 10

Mean = np = 10 * 0.50 = 5

Standard Deviation = = = 1.581139

6.

Probability of success, p = 0.10

Sample size, n = 10

Mean = np = 10 * 0.10 = 1

Standard Deviation = = = 0.9486833

When the probability of success decreases keeping the sample size constant, the mean decreases. As the probability of success decreases which leads to decreases the observed value and hence the expected value (mean) also decreases. The standard deviation of binomial random variable is highest at p = 0.5. As, the probability of success deviates from 0.5, the standard deviation decreases.

7.

Probability of success, p = 0.90

Sample size, n = 10

Mean = np = 10 * 0.90 = 9

Standard Deviation = = = 0.9486833

When the probability of success increases keeping the sample size constant, the mean increases. As the probability of success increases which leads to increases the observed value and hence the expected value (mean) also increases. The standard deviation of binomial random variable is highest at p = 0.5. As, the deviation of p = 0.9 from 0.5 is same as for p = 0.1, the standard deviation will be equal for p = 0.1 and 0.9.