Question

Nine West sells women's flip flops in eight different colors. Despite
the available selection, 50% of all filp flops purchased are white.
Suppose 30 buyers are selected at random.
a. Find the mean, variance, and standard deviation of the number of
buyers who purchase white flip flops.
b. Find the probability that the number of white flip flops purchased
will be within two standard deviations of the mean. Compare this
with the predicted result from Chebyshev's Rule.
c. Suppose two groups of 30 customers are independently selected.
What is the probability of at least one group having exactly 15
people who buy white flip flops.

Answer 1

Let x be the number of buyers who purchase white flip flops.

We are given n = 30, p = 0.50 , q = 1 - p =1- 0.5 = 0.5

x follows binomial distribution with n = 30 and p = 0.5

a) Mean = n* p = 0.5*30 = 15

Variance = n*p*q = 30*0.5*0.5 = 7.5

Standard deviation = = = 2.7386

b) Mean -(2*standard deviation ) = 15 - (2*2.7386 ) = 9.52 ~ 10

Mean +(2*standard deviation ) = 15 + (2*2.7386 ) = 20.48 ~ 20

We have to find P( 10 ≤ x ≤ 20 )

We can find binomial probabilities using excel function =BINOM.DIST(x,n,p,FALSE)

So probability that the number of white flip flops purchased will be within two standard deviations of the mean = 0.9573 or 95.73%

According to Chebyshev's Rule.

at least 75% of the data values must be within 2 standard deviations of the mean.

So the % of theoretical probability and Chebyshev's Rule are not same.

c)  probability of one group having exactly 15 people who buy white flip flops. P( x = 15 ) = 0.1445

So  probability of one group do not have exactly 15 people who buy white flip flops. = 1- 0.1445 = 0.8555

P( at least one group have exactly 15 people ) = 1 -P( both group do not have exactly 15 people )

= 1- ( 0.8555 *0.8555 )

= 1 - 0.7319

= 0.2681