Question

Consider the following table:

Defects in batch	Probability
0	0.30
1	0.28
2	0.21
3	0.09
4	0.08
5	0.04

Find the standard deviation of this variable.

0.67

1.49

1.99

1.41

Question 5

(CO 4) Twenty-two percent of US teens have heard of a fax machine. You randomly select 12 US teens. Find the probability that the number of these selected teens that have heard of a fax machine is exactly six (first answer listed below). Find the probability that the number is more than 8 (second answer listed below).

0.993, 0.024

0.993, 0.000

0.024, 0.001

0.024, 0.000

Answer 1

Solution :

From given probability distribution table,

= X * P(X)

= 0 * 0.30 + 1 * 0.28 + 2 * 0.21 + 3 * 0.09 + 4 * 0.08 + 5 * 0.04

= 0 + 0.28 + 0.42 + 0.27 + 0.32 + 0.20

= 1.49

variance = X² * P(X) - ²

= [0² * 0.30 + 1² * 0.28 + 2² * 0.21 + 3² * 0.09 + 4² * 0.08 + 5² * 0.04] - 1.49²

= 0 + 0.28 + 0.84 + 0.81 + 1.28 + 1 - 2.2201

= 1.99

variance = 1.99

standard deviation = 1.99 = 1.41

The standard deviation of this variable is 1.41

5)

Solution

Given that ,

p = 22% = 0.22

1 - p = 1 - 0.22 = 0.78

n = 12

Using binomial probability formula ,

P(X = x) = ((n! / x! (n - x)!) * p^x * (1 - p)^{n - x}

1)

P(X = 6) = ((12! / 6! (12 - 6)!) * 0.22⁶ * (0.78)^{12 - 6}

=  ((12! / 6! (6)!) * 0.22⁶ * (0.78)⁶

=  0.024

Probability = 0.024

2)

P(X > 8) = P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)

= ((12! / 9! (3)!) * 0.22⁹ * (0.78)³ + ((12! / 10! (2)!) * 0.22¹⁰ * (0.78)² +

((12! / 11! (1)!) * 0.22¹¹ * (0.78)¹ + ((12! / 12! (0)!) * 0.22¹² * (0.78)⁰

= 0.000

Probability = 0.000

Answer : 0.024, 0.000