Question

In a large restaurant an average 60% customers ask for water with their meal. A random sample of 10 customers is selected. Find the probability that,

(a) exactly 6 ask for water with their meal(2 points)
(b) less than 9 ask for water with their meal.(2 points)
(c) at least 3 ask for water with their meal.(2 points)
(d) Find the mean and the standard deviation. (3 points)

Answer 1

a)
`P(6)=0.25`

b)
`p(x<g)=0.9537`

c)
`p(x\geq3)=0.9878`

d)
`\sigma=√(2.4)=1.5492`

Explanation:

From the question we are told that:

Population percentage
`p_\%=\60%`

Sample size
`n=10`

Let x =customers ask for water

Let y =customers dose not ask for water with their meal

Generally the equation for y is mathematically given by

`y=1-p_\%\\y=1-0.60\\y=0.40`

Generally the equation for pmf p(x) is mathematically given by

`P(x)=10C_x (0..6)^x(0.4)^(10-x)`

a)

Generally the probability that exactly 6 ask for water is mathematically given by

`P(x)6=10C_6 (0..6)^6(0.4)^(10-6)`

`P(6)=0.25`

b)

Generally the probability that less than 9 ask for water with meal is mathematically given by

`p(x<g)=1-p(x>g)`

`p(x<g)=1(p(9))+p(10)`

`p(x<g)=1-(10_C_9 (0..6)^9(0.4)^(10-9)+10_C_10 (0..6)^10(0.4)^(10-10))\\p(x<g)=1-0.0463`

`p(x<g)=0.9537`

c)

Generally the probability that at least 3 ask for water with meal is mathematically given by

`p(x\geq3)=1-p(x<3)`

`p(x\geq3)=1-[p(0)+p(1)+p(2)]`

`p(x\geq3)=1-[0.00001+0.0015+0.0106]`

`p(x\geq3)=1-[0.0122]`

`p(x\geq3)=0.9878`

d)

Generally the mean and standard deviation of sample size is mathematically given by

Mean

`\=x=np=10(0.6)=6`

Standard deviation

`v(x)=npq=10(0.6)(0.4)=2.4`

`\sigma=√(2.4)=1.5492`