Question

A running shoe store mounts a new promotional campaign. Purchasers of new shoes may, if dissatisfied for any reason, return them within 7 days of purchase and receive a full refund. The cost to the dealer of such a refund is $105. The dealer estimates that with an 11% probability each purchaser, for whatever reason, will return their pair of shoes and obtain refunds. Suppose that 120 pairs of shoes are purchased during the campaign period. Clearly the exact number of purchasers who return their product is unknown. Answer the following questions:

a. Describe the probability distribution of the number of pairs of shoes that will be returned for refunds.
b. What is the mean and standard deviation of the number of pairs of shoes that will be returned for refunds.
c. What is the mean and standard deviation of the total refund costs that will accrue as a result of these refunds.

Answer 1

a

`P(K = k ) = \  ^nC_x * p^k  *  (1-p)^(n-k)`

Here k is the random number of number of pairs of shoes that will be returned for refund, which can be k = 0,1 ,2,3,4 ..., 120

b

The mean is
`E(K) =  np =  13.2`

The standard deviation is
`\sigma =  3.43`

c

The mean is
`E(C) =   E(105 K) =  1386`

The standard deviation is
`\sigma  =  381.48`

Explanation:

From the question we are told that

The cost to dealer for each refund is
`C =  \$105`

The probability that a shoe will refunded is
`P(x) =  0.11`

The number of shoes purchased is n = 120 pairs

Generally the probability distribution of the number of pairs of shoes that will be returned for refunds is

`P(K = k ) = \  ^nC_x * p^k  *  (1-p)^(n-k)`

Here k is the random number of number of pairs of shoes that will be returned for refund, which can be k = 0,1 ,2,3,4 ..., 120

Generally the mean is mathematically represented as

`E(K) =  np =  120 * 0.11`

=>
`E(K) =  np =  13.2`

Generally the standard deviation is mathematically represented as

`\sigma  =  √( n*  p  (1-p))`

=>
`\sigma  =  √( 120* 0.11  (1-0.11))`

=>
`\sigma =  3.43`

Generally the total refund cost is mathematically represented as

`C =  105 K`

Here K denotes the number of shoes returned

Generally the mean of the total refund cost is mathematically represented as

`E(C) =   E(105 K) =  105 E(K)`

=>
`E(C) =   E(105 K) =  105 * 13.2`

=>
`E(C) =   E(105 K) =  1386`

Generally the variance of the total refund cost is mathematically represented as

`Var(K) =  105^2 *  E(K)`

=>
`Var(K) =  105^2 *  13.2`

=>
`Var(K) =  145530`

=>
`\sigma =  √(145530 )`

=>
`\sigma  =  381.48`