Question

Minitab may be used, attach or include your output if you use Minitab to find answers.

1) Shower temperature at the Spokane Club showers is regulated automatically. The heater kicks in when the temperature falls to 980F and shuts off when the temperature reaches 1080 Water temperature then falls slowly until the heater kicks in again. At a given moment, the water temperature is a uniformly distributed random variable U(98,108). Show your work. (15 points)

Find the mean temperature.

Find the standard deviation of the temperature.

Find the probability that the temperature is less than 100 degrees.

Find the probability that the temperature is less than 102 degrees or more than 104 degrees.

Find the 75th percentile for water temperature.

2) The weekly demand for Rosa’s pizzas on a Friday night is a normally distributed random variable with mean 235 and standard deviation 10. Show your work. (15 points)

Find the probability that demand exceeds 220.

Find the probability that demand is between 230 and 245.

Find the probability that demand is less than 225 or more than 250.

Find the number of pizzas demanded at the first Quartile.

Suppose the store wants to make sure they are able to meet demand 90% of the time, how many pizzas should they bake?

Answer 1

1)

a) mean temperature=(b+a)/2=(108+98)/2=103

b) standard deviation=(b-a)/sqrt(12)=(108-98)/sqrt(12)=2.887

c) probability that the temperature is less than 100 degrees=P(X<100)=(x-a)/(b-a)=(100-98)/(108-98)=2/10=0.2

d)P(X<102)+P(X>104)=1-P(102<X<104)=1-(104-102)/10=1-0.2=0.8

e)

75 th percentile =a+0.75*(b-a)=98+0.75*10=105.5

2)

a)

for normal distribution z score =(X-μ)/σx
here mean=       μ=	235
std deviation   =σ=	10.0000

probability that demand exceeds 220:

probability =

P(X>220)

=

P(Z>-1.5)=

1-P(Z<-1.5)=

1-0.0668=

0.9332

b)

probability that demand is between 230 and 245:

probability =

P(230<X<245)

=

P(-0.5<Z<1)=

0.8413-0.3085=

0.5328

c)

probability that demand is less than 225 or more than 250:

P(X<225)+P(X>250)=1-(225<X<250)=1-(-1<Z<1.5)=1-(0.9332-0.1587)=0.2255

d)

for 25th percentile critical value of z=		-0.67
therefore corresponding value=mean+z*std deviation=			228.3

e)

for 90th percentile critical value of z=		1.28
therefore corresponding value=mean+z*std deviation=			247.8