Answer:
the answer is below
Step-by-step explanation:
The z score is used to calculate by how many standard deviations the raw score is above or below the mean. The z score is given as:
[tex]z=\frac{x-\mu}{\sigma}\\\\\mu=mean,\sigma=standard\ deviation[/tex]
1) For x = 3
[tex]z=\frac{x-\mu}{\sigma}=\frac{3-4}{2}=-0.5[/tex]
For x = 6
[tex]z=\frac{x-\mu}{\sigma}=\frac{6-4}{2}=1[/tex]
P(3 ≤ x ≤ 6) = P(-0.5 ≤ z ≤ 1) = P(z < 1) - P(z < -0.5) = 0.8413 - 0.3085 = 0.5328
2) For x = 50
[tex]z=\frac{x-\mu}{\sigma}=\frac{50-40}{15}=0.67[/tex]
For x = 70
[tex]z=\frac{x-\mu}{\sigma}=\frac{70-40}{15}=2[/tex]
P(50 ≤ x ≤ 70) = P(0.67 ≤ z ≤ 2) = P(z < 2) - P(z < 0.67) = 0.9772 - 0.7486 = 0.2286
3) For x = 8
[tex]z=\frac{x-\mu}{\sigma}=\frac{8-15}{3.2}=-2.19[/tex]
For x = 12
[tex]z=\frac{x-\mu}{\sigma}=\frac{12-15}{3.2}=-0.94[/tex]
P(8 ≤ x ≤ 12) = P(-2.19 ≤ z ≤ -0.94) = P(z < -0.94) - P(z < -2.19) = 0.1736 - 0.0143 = 0.1593
4) For x = 30
[tex]z=\frac{x-\mu}{\sigma}=\frac{30-20}{3.4}=2.94[/tex]
P(x ≥ 30) = P(z ≥ 2.94) = 1 - P(z < 2.94) = 1 - 0.9984 = 0.0016
5) x = 90
[tex]z=\frac{x-\mu}{\sigma}=\frac{90-100}{15}=-0.67[/tex]
P(x ≥ 90) = P(z ≥ -0.67) = 1 - P(z < -0.67) = 1 - 0.2514 = 0.7486
6) For x = 10
[tex]z=\frac{x-\mu}{\sigma}=\frac{10-15}{4}=-1.25[/tex]
For x = 20
[tex]z=\frac{x-\mu}{\sigma}=\frac{20-15}{4}=1.25[/tex]
P(10 ≤ x ≤ 20) = P(-1.25 ≤ z ≤ 1.25) = P(z < 1.25) - P(z < -1.25) = 0.8944 - 0.1056 = 0.7888
7) For x = 7
[tex]z=\frac{x-\mu}{\sigma}=\frac{7-5}{1.2}=1.67[/tex]
For x = 9
[tex]z=\frac{x-\mu}{\sigma}=\frac{9-5}{1.2}=3.33[/tex]
P(7 ≤ x ≤ 9) = P(1.67 ≤ z ≤ 3.33) = P(z < 3.33) - P(z < 1.67) = 0.9996 - 0.9525 = 0.0471
8) For x = 40
[tex]z=\frac{x-\mu}{\sigma}=\frac{40-50}{15}=-0.67[/tex]
For x = 47
[tex]z=\frac{x-\mu}{\sigma}=\frac{47-50}{15}=-0.2[/tex]
P(40 ≤ x ≤ 47) = P(-0.67 ≤ z ≤ -0.2) = P(z < -0.2) - P(z < -0.67) = 0.4207 - 0.2514 = 0.1693
9) x = 120
[tex]z=\frac{x-\mu}{\sigma}=\frac{120-10}{15}=7.33[/tex]
P(x ≥ 120) = P(z ≥ 7.33) = 1 - P(z < 7.33) = 1 - 0.9999 = 0.001
10) x = 2
[tex]z=\frac{x-\mu}{\sigma}=\frac{2-3}{0.25}=-4[/tex]
P(x ≥ 2) = P(z ≥ -4) = 1 - P(z < -4) = 1 - 0.0001 = 0.999
