Respuesta :
Answer:
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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:
[tex]\mu = 63, \sigma = 18[/tex]
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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{60 - 63}{18}[/tex]
[tex]Z = -0.17[/tex]
[tex]Z = -0.17[/tex] 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.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.645 = \frac{X - 63}{18}[/tex]
[tex]X - 63 = -1.645*18[/tex]
[tex]X = 33.39[/tex]
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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.675 = \frac{X - 63}{18}[/tex]
[tex]X - 63 = -0.675*18[/tex]
[tex]X = 50.85[/tex]
Q1: 50.85 minutes
Q3 = 75th percentile
Value of X when Z has a pvalue of 0.75. So X when Z = 0.675
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.675 = \frac{X - 63}{18}[/tex]
[tex]X - 63 = 0.675*18[/tex]
[tex]X = 75.15[/tex]
Q3: 75.15 minutes
IQR = Q3 - Q1:
75.15 - 50.85 = 24.3
IQR: 24.3 minutes
