Respuesta :
Answer:
81.74% probability that the sum of the 95 wait times you observed is between 670 and 796
Step-by-step explanation:
To solve this question, the uniform probability distribution and the normal probability distribution must be understood.
Uniform distribution:
A distribution is called uniform if each outcome has the same probability of happening.
The distribution has two bounds, a and b.
Its mean is given by:
[tex]M = \frac{b - a}{2}[/tex]
Its standard deviation is given by:
[tex]S = \sqrt{\frac{(b-a)^2}{12}}[/tex]
Normal distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
n instances of the uniform distribution can be approximated to the normal with [tex]\mu = nM[/tex], [tex]\sigma = S\sqrt{n}[/tex]
Uniformly distributed over the interval [0,15].
This means that:
[tex]M = \frac{15 - 0}{2} = 7.5[/tex]
[tex]S = \sqrt{\frac{(15-0)^2}{12}} = 4.33[/tex]
95 trains
[tex]n = 95[/tex], so:
[tex]\mu = 95M = 95*7.5 = 712.5[/tex]
[tex]\sigma = S\sqrt{n} = 4.33\sqrt{95} = 42.2[/tex]
What is the approximate probability (to 2 decimal places) that the sum of the 95 wait times you observed is between 670 and 796?
This is the pvalue of Z when X = 796 subtracted by the pvalue of Z when X = 670. So
X = 796
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{796 - 712.5}{42.2}[/tex]
[tex]Z = 1.98[/tex]
[tex]Z = 1.98[/tex] has a pvalue of 0.9761
X = 670
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{670 - 712.5}{42.2}[/tex]
[tex]Z = -1[/tex]
[tex]Z = -1[/tex] has a pvalue of 0.1587
0.9761 - 0.1587 = 0.8174
81.74% probability that the sum of the 95 wait times you observed is between 670 and 796
