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 OvernightPlan, 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

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 OvernightPlan, 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.

[tex]E(x)=\sum x\times P(x)[/tex]

[tex]E(x)=0\times 0.1+1\times 0.2+2\times 0.7[/tex]

[tex]E(x)=0+0.2+1.4[/tex]

[tex]E(x)=1.6[/tex]

E(x)= 1.6 nights.

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

[tex]\sigma=\sqrt{\text{Variance}}[/tex]

[tex]\text{Variance}=E(X^2)-E(X)^2[/tex]

[tex]E(X)^2=(1.6)^2=2.56[/tex]

[tex]E(X^2)=0^2\times 0.1+1^2\times 0.2+2^2\times 0.7[/tex]

[tex]E(X^2)=0+0.2+2.8[/tex]

[tex]E(X^2)=3[/tex]

[tex]\text{Variance}=3-2.56[/tex]

[tex]\text{Variance}=0.44[/tex]

[tex]\sigma=\sqrt{0.44}[/tex]

[tex]\sigma=0.663[/tex]

[tex]\sigma=0.663[/tex] of nights.