Answer:
(1) The probability that the time to deliver a pizza is at least 32 minutes is 0.70.
(2a) The percentage of results more than 45 is 79.67%.
(2b) The percentage of results less than 85 is 91.77%.
(2c) The percentage of results are between 75 and 90 is 15.58%.
(2d) The percentage of results outside the healthy range 20 to 100 is 2.64%.
Step-by-step explanation:
(1)
Let <em>Y</em> = the time taken to deliver a pizza.
The random variable <em>Y</em> follows a Uniform distribution, U (20, 60).
The probability distribution function of a Uniform distribution is:
![f(x)=\left \{ {{\frac{1}{b-a};\ x\in [a, b] } \atop {0};\ otherwise} \right.](https://tex.z-dn.net/?f=f%28x%29%3D%5Cleft%20%5C%7B%20%7B%7B%5Cfrac%7B1%7D%7Bb-a%7D%3B%5C%20x%5Cin%20%5Ba%2C%20b%5D%20%7D%20%5Catop%20%7B0%7D%3B%5C%20otherwise%7D%20%5Cright.)
Compute the probability that the time to deliver a pizza is at least 32 minutes as follows:
![P(Y\geq 32)=\int\limits^{60}_{32} {\frac{1}{b-a} } \, dx \\=\frac{1}{60-20} \int\limits^{60}_{32} {1 } \, dx\\=\frac{1}{40}\times[x]^{60}_{32}\\=\frac{1}{40}\times[60-32]\\=0.70](https://tex.z-dn.net/?f=P%28Y%5Cgeq%2032%29%3D%5Cint%5Climits%5E%7B60%7D_%7B32%7D%20%7B%5Cfrac%7B1%7D%7Bb-a%7D%20%7D%20%5C%2C%20dx%20%5C%5C%3D%5Cfrac%7B1%7D%7B60-20%7D%20%5Cint%5Climits%5E%7B60%7D_%7B32%7D%20%7B1%20%7D%20%5C%2C%20dx%5C%5C%3D%5Cfrac%7B1%7D%7B40%7D%5Ctimes%5Bx%5D%5E%7B60%7D_%7B32%7D%5C%5C%3D%5Cfrac%7B1%7D%7B40%7D%5Ctimes%5B60-32%5D%5C%5C%3D0.70)
Thus, the probability that the time to deliver a pizza is at least 32 minutes is 0.70.
(2)
Let <em>X</em> = results of a certain blood test.
It is provided that the random variable <em>X</em> follows a Normal distribution with parameters 
and 
.
The probabilities of a Normal distribution are computed by converting the raw scores to <em>z</em>-scores.
The <em>z</em>-scores follows a Standard normal distribution, N (0, 1).
(a)
Compute the probability that the results are more than 45 as follows:
The percentage of results more than 45 is: 
Thus, the percentage of results more than 45 is 79.67%.
(b)
Compute the probability that the results are less than 85 as follows:
The percentage of results less than 85 is: 
Thus, the percentage of results less than 85 is 91.77%.
(c)
Compute the probability that the results are between 75 and 90 as follows:
The percentage of results are between 75 and 90 is: 
Thus, the percentage of results are between 75 and 90 is 15.58%.
(d)
Compute the probability that the results are between 20 and 100 as follows:
Then the probability that the results outside the range 20 to 100 is: 
.
The percentage of results outside the range 20 to 100 is: 
Thus, the percentage of results outside the healthy range 20 to 100 is 2.64%.
