Question

A marketing agency has developed three vacation packages to promote a timeshare plan at a new resort. They estimate that 10​% of potential customers will choose the Day​ Plan, which does not include overnight​ accommodations; 20​% will choose the Overnight​Plan, which includes one night at the​ resort; and 70​% will choose the Weekend​ Plan, which includes two nights. Complete parts a and b below.

a) Find the expected value of the number of nights potential customers will need.

​E(X)= _____ of nights

​b) Find the standard deviation of the number of nights potential customers will need.

σ = _________ of nights

Answer 1

a) E(x)= 1.6 nights.

b)
`\sigma=0.663` of nights.

Explanation:

Given : A marketing agency has developed three vacation packages to promote a timeshare plan at a new resort. They estimate that 10​% of potential customers will choose the Day​ Plan, which does not include overnight​ accommodations; 20​% will choose the Overnight​Plan, which includes one night at the​ resort; and 70​% will choose the Weekend​ Plan, which includes two nights.

The table form is

x(nights) 0 1 2

P(x) 0.1 0.2 0.7

a) Find the expected value of the number of nights potential customers will need.

`E(x)=\sum x* P(x)`

`E(x)=0* 0.1+1* 0.2+2* 0.7`

`E(x)=0+0.2+1.4`

`E(x)=1.6`

E(x)= 1.6 nights.

​b) Find the standard deviation of the number of nights potential customers will need.

`\sigma=\sqrt{\text{Variance}}`

`\text{Variance}=E(X^2)-E(X)^2`

`E(X)^2=(1.6)^2=2.56`

`E(X^2)=0^2* 0.1+1^2* 0.2+2^2* 0.7`

`E(X^2)=0+0.2+2.8`

`E(X^2)=3`

`\text{Variance}=3-2.56`

`\text{Variance}=0.44`

`\sigma=√(0.44)`

`\sigma=0.663`

`\sigma=0.663` of nights.