The amount of time that people spend at Grover Hot Springs is normally distributed with a mean of 63 minutes and a standard deviation of 18 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 two decimal places. A. X ~ N( , )

B. Find the probability that a randomly selected person at the hot springs stays longer then an hour (60 minutes).

C. The park service is considering offering a discount for the 5% 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.

a) X ~ N(63,18)

b) 0.5675 = 56.75%

c) 33.39 minutes

d) Q1: 50.85 minutes

Q3: 75.15 minutes

IQR: 24.3 minutes

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

A. X ~ N(63,18)

The mean and then the standard deviation.

B. Find the probability that a randomly selected person at the hot springs stays longer then an hour (60 minutes).

This is 1 subtracted by the pvalue of Z when X = 60. So

has a pvalue of 0.4325

1 - 0.4325 = 0.5675

C. The park service is considering offering a discount for the 5% 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?

5th percentile, which is the value of X when Z has a pvalue of 0.05. So it is X when Z = -1.645.

33.39 minutes

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

Q1 = 25th percentile.

Value of X when Z has a pvalue of 0.25. So X when Z = -0.675

Q1: 50.85 minutes

Q3 = 75th percentile

Value of X when Z has a pvalue of 0.75. So X when Z = 0.675

Q3: 75.15 minutes

IQR = Q3 - Q1:

75.15 - 50.85 = 24.3

IQR: 24.3 minutes

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step-by-step explanation: