Question

The amount of time that people spend at Grover Hot Springs is normally distributed with a mean of 80 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 89 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

Solution :

Given that ,

mean = = 80

standard deviation = = 14

a) The distribution of x is normal X N ( 80, 14 )

b) P(x > 89) = 1 - p( x< 89)

=1- p P[(x - ) / < (89 - 80) / 14]

=1- P(z < 0.64)

Using z table,

= 1 - 0.7389

= 0.2611

c) Using standard normal table,

P(Z < z) = 4%

= P(Z < z) = 0.04  

= P(Z < -1.75) = 0.04

z = -1.75

Using z-score formula,

x = z * +

x = -1.75 * 14 + 80

x = 55.5 minutes.

d) Using standard normal table,

The z dist'n First quartile is,

P(Z < z) = 25%

= P(Z < z) = 0.25  

= P(Z < -0.67 ) = 0.25

z = -0.67

Using z-score formula,

x = z * +

x = -0.67 * 14 + 80

x = 70.62

First quartile =Q1 = 70.62 minutes

The z dist'n Third quartile is,

P(Z < z) = 75%

= P(Z < z) = 0.75  

= P(Z < 0.67 ) = 0.75

z = 0.67

Using z-score formula,

x = z * +

x = 0.67 * 14 + 80

x = 89.38

Third quartile =Q3 = 89.38 minutes

IQR = Q3 - Q1

IQR = 89.38 - 70.62

IQR = 18.76 minutes