Standard Deviation

Question

Consider a nationally recognized standardized college admissions test with scores that are normally distributed, centered at a mean of 21, and have a standard deviation of 4.5. Use the 68-95-99.7 Rule to find the percentage of scores in each of the following scenarios

 

1. Percentage of scores less than 16.5:___%

 

 
2. Percentage of scores greater than 12:___%

 

 

3. Percentage of scores between 21 and 25.5: ___%

 

 

4. Percentage of scores between 12 and 25.5: ___%

Answer 1

First let's calculate some probabilities for z-scores:

 

P(|z| > 1) = 1 - P(|z| ≤ 1) = 1 - 0.68 = 0.32

 

P(z < -1) = P(z > 1) = 0.32 / 2 = 0.16

 

P(|z| > 2) = 1 - P(|z| ≤ 2) = 1 - 0.95 = 0.05

P(z < -2) = P(z > 2) = 0.05 / 2 = 0.025

 

P(|z| > 3) = 1 - P(|z| ≤ 3) = 1 - 0.997 = 0.003

P(z < -1) = P(z > 1) = 0.003 / 2 = 0.0015

 

Now for the problems:

 

1. z = (16.5 - 21)/4.5 = -1

 

P(z < -1) = 16%

 

2. z = (12 - 21)/4.5 = -2

P(z > -2) = (100 - 2.5)% = 97.5%

 

3. z_L = (21 - 21) / 4.5 = 0; z_H = (25.5 - 21) / 4.5 = 1

P(0 < z < 1) = (68%) / 2 = 34%

3. z_L = (12 - 21) / 4.5 = -2; z_H = (25.5 - 21) / 4.5 = 1

P(-2 < z < 1) = (95%) / 2 + (68%) / 2 = (47.5 + 34)% = 81.5%