Question

The amount of time that people spend at Grover Hot Springs is normally distributed with a mean of 74 minutes and a standard deviation of 14 minutes. Suppose one person at the hot springs is randomly chosen. Let X = the amount of time that person spent at Grover Hot Springs . Round all answers to 4 decimal places where possible.

a. What is the distribution of X? X ~ N( , )

b. Find the probability that a randomly selected person at the hot springs stays longer then 98 minutes.

c. The park service is considering offering a discount for the 4% of their patrons who spend the least time at the hot springs. What is the longest amount of time a patron can spend at the hot springs and still receive the discount? minutes.

d. Find the Inter Quartile Range (IQR) for time spent at the hot springs.

Q1: minutes

Q3: minutes

IQR: minutes

Answer 1

Answer)

As the data is normally distributed we can use standard normal z table to estimate the answers

Z = (x-mean)/s.d

Given mean = 74

S.d = 14

A)

X ~ N(74, 14)

B)

P(x>98)

Z = 1.71

From z table, P(z>1.71) = 0.0436

C)

P(z<-1.75) = 4%

So,

-1.75 = (x - 74)/14

X = 49.5

D)

Interquartile range = Q3 - Q1

We know that below Q3, 75% lies

From z table, P(z<0.67) = 75%

0.67 = (Q3 - 74)/14

Q3 = 83.38

We know that below Q1, 25% lies

P(z<-0.67) = 25%

-0.67 = (Q1 - 74)/14

Q1 = 64.62

IQR = 83.38 - 64.62 = 18.76