Answer:
The answer is below
Explanation:
A normal model is represented as (μ, σ). Therefore for (1.5, 0.18), the mean (μ) = 1.5 and the standard deviation (σ) = 0.18
The z score shows by how many standard deviations the raw score is above or below the mean. It is given as:
![z=\frac{x-\mu}{\sigma}](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D)
a) For x < 1.35 s
![z=\frac{x-\mu}{\sigma}\\\\z=\frac{1.35-1.5}{0.18}=-0.83](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%5C%5C%5C%5Cz%3D%5Cfrac%7B1.35-1.5%7D%7B0.18%7D%3D-0.83)
From the normal distribution table, the percent of drivers have a reaction time less than 1.35 seconds = P(x < 1.35) = P(z < -0.83) = 0.2033 = 20.33%
b) For x > 1.9 s
![z=\frac{x-\mu}{\sigma}\\\\z=\frac{1.9-1.5}{0.18}=2.22](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%5C%5C%5C%5Cz%3D%5Cfrac%7B1.9-1.5%7D%7B0.18%7D%3D2.22)
From the normal distribution table, the percent of drivers have a reaction time greater than 1.9 seconds = P(x > 1.9) = P(z > 2.22) = 1 - P(z<2.22) = 1 - 0.9868 = 0.0132 = 1.32%
c) For x = 1.45
![z=\frac{x-\mu}{\sigma}\\\\z=\frac{1.45-1.5}{0.18}=-0.28](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%5C%5C%5C%5Cz%3D%5Cfrac%7B1.45-1.5%7D%7B0.18%7D%3D-0.28)
For x = 1.75
![z=\frac{x-\mu}{\sigma}\\\\z=\frac{1.75-1.5}{0.18}=1.39](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%5C%5C%5C%5Cz%3D%5Cfrac%7B1.75-1.5%7D%7B0.18%7D%3D1.39)
From the normal distribution table, P(1.45 < x < 1.75) = P(-0.28 < z < 1.39) = P(z < 1.39) - P(z< - 0.28) = 0.9177 - 0.3897 = 0.528 = 52.8%
d) A percentage of 10% corresponds to a z score of -1.28
![z=\frac{x-\mu}{\sigma}\\\\-1.28=\frac{x-1.5}{0.18}\\\\x-1.5=-0.2034\\\\x=1.27](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%5C%5C%5C%5C-1.28%3D%5Cfrac%7Bx-1.5%7D%7B0.18%7D%5C%5C%5C%5Cx-1.5%3D-0.2034%5C%5C%5C%5Cx%3D1.27)
e) P(z < z1) - P(z< -z1) = 60%
P(z < z1) - P(z< -z1) = 0.6
P(z < -z1) = 1 - P(z < z1)
P(z<z1) - (1 - P(z < z1)) = 0.6
2P(z<z1) - 1= 0.6
2P(z<z1) = 1.6
P(z<z1) = 0.8
From the z table, z1 = 0.85
![0.85=\frac{x-1.5}{0.18}and-0.85=\frac{x -1.5}{0.18} \\\\x=1.65 \ and\ x=1.35](https://tex.z-dn.net/?f=0.85%3D%5Cfrac%7Bx-1.5%7D%7B0.18%7Dand-0.85%3D%5Cfrac%7Bx%20-1.5%7D%7B0.18%7D%20%20%5C%5C%5C%5Cx%3D1.65%20%5C%20and%5C%20x%3D1.35)
The reaction time between 1.35 and 1.65 seconds