Question

# The amount of time that people spend at Grover Hot Springs is normally distributed with a...

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

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