Question

Eighty-five percent of the employees at the Smucker Company plant on Colorado Road have their bimonthly wages sent directly to their bank by electronic funds transfer. This is called direct deposit. Suppose we select a random sample of twelve employees.
a) What is the probability that more than seven of them receive their pay via direct deposit?
b) What is the probability that at least six of them receive their pay via direct deposit?

c) List the probabilities for x =0-12. And find the mean and standard deviation of this distribution using the list of probabilities (x=0-12).

Answer 1

X: Number of employees receive their pay via direct deposit

P:- Percent of  employee  receive their pay via direct deposit

n:- Total number of employees sampled.

P=0.85 ; n = 12

X follows Binomial( n= 12, P= 0.85)

The probability mass function for X is

Let's first solve Q.c) it will help to solve first two

C)

n= 12 , P=0.85

Sr.No.	X	12cX	(0.85)^x	(0.15)^(12-x)	P(x)
1	0	1	1	1.3E-10	1.3E-10
2	1	12	0.85	8.6E-10	8.8E-09
3	2	66	0.7225	5.8E-09	2.7E-07
4	3	220	0.614125	3.8E-08	5.2E-06
5	4	495	0.522006	2.6E-07	6.6E-05
6	5	792	0.443705	1.7E-06	0.0006
7	6	924	0.37715	1.1E-05	0.0040
8	7	792	0.320577	7.6E-05	0.0193
9	8	495	0.272491	0.0005	0.0683
10	9	220	0.231617	0.0034	0.1720
11	10	66	0.196874	0.0225	0.2924
12	11	12	0.167343	0.15	0.3012
13	12	1	0.142242	1	0.1422
Total					1.00

Above is the table for the probability of x for 0 to 12.

Sr.No.	X	P(x)	X.P(x)	X^2	(X^2)*P(x)
1	0	1.3E-10	0.00	0	0.00
2	1	8.8E-09	8.8E-09	1	8.8E-09
3	2	2.7E-07	5.5E-07	4	1.1E-06
4	3	5.2E-06	1.6E-05	9	4.7E-05
5	4	6.6E-05	0.0003	16	0.0011
6	5	0.0006	0.0030	25	0.0150
7	6	0.0040	0.0238	36	0.1429
8	7	0.0193	0.1350	49	0.9447
9	8	0.0683	0.5463	64	4.3702
10	9	0.1720	1.5478	81	13.9300
11	10	0.2924	2.9236	100	29.2358
12	11	0.3012	3.3134	121	36.4474
13	12	0.1422	1.7069	144	20.4828
Total		1.00	10.20		105.57