a) A certain machine produces bulbs and were examined for the number of defectives in a sample of 14. The following table shows the distribution the number of defective items contained in 139 samples:

Number of Defectives in a Sample of 14

0

1

2

3

4

5

6

7

8

Number of Samples

7

8

20

21

35

30

12

1

5

i) Fit a binomial distribution and find the expected frequencies if the chance of machine being defective is 0.5

ii) Find the probability of that number of defectives are at least 3.

iii) Find the mean and standard deviation of the number of defectives.

b) A new automated production process has an average of 1.5 breakdowns per day. Because of the cost associated with a breakdown, management is concerned about the possibility of having three or more breakdowns during a day. Assume that breakdowns occur randomly, that the probability of a breakdown is the same for any two time intervals of equal length, and that breakdowns in one period are independent of breakdowns in other periods. What is the probability of having three or more breakdowns during a day?