Let's solve each of the given parts step-by-step using the properties of the normal distribution.
Given:
- Mean (μ) = 90
- Standard Deviation (σ) = 15
### a. Greater than 90
The mean of the distribution is 90. As per the properties of a normal distribution, 50% of the values lie above the mean.
- The percentage of scores greater than 90 is 50%.
### c. Less than 75
To find the percentage of scores less than 75, we calculate the Z-score for 75:
[tex]\[ Z = \frac{X - \mu}{\sigma} = \frac{75 - 90}{15} = -1 \][/tex]
A Z-score of -1 corresponds to the lower 15.9% of the distribution.
- The percentage of scores less than 75 is 15.9%.
### e. Less than 60
To find the percentage of scores less than 60, we calculate the Z-score for 60:
[tex]\[ Z = \frac{X - \mu}{\sigma} = \frac{60 - 90}{15} = -2 \][/tex]
A Z-score of -2 corresponds to the lower 2.3% of the distribution.
- The percentage of scores less than 60 is 2.3%.
### g. Greater than 75
Using the result from part c, we know that 15.9% of the distribution is less than 75.
To find the percentage greater than 75:
[tex]\[ \text{Percentage greater than 75} = 100\% - 15.9\% = 84.1\% \][/tex]
- The percentage of scores greater than 75 is 84.1%.
### b. Greater than 105
To find the percentage of scores greater than 105, we calculate the Z-score for 105:
[tex]\[ Z = \frac{X - \mu}{\sigma} = \frac{105 - 90}{15} = 1 \][/tex]
A Z-score of 1 means that 84.1% of the values lie below 105, so:
[tex]\[ \text{Percentage greater than 105} = 100\% - 84.1\% = 15.9\% \][/tex]
- The percentage of scores greater than 105 is 15.9%.
### d. Less than 120
To find the percentage of scores less than 120, we calculate the Z-score for 120:
[tex]\[ Z = \frac{X - \mu}{\sigma} = \frac{120 - 90}{15} = 2 \][/tex]
A Z-score of 2 corresponds to the lower 97.7% of the distribution.
- The percentage of scores less than 120 is 97.7%.
### f. Less than 105
To find the percentage of scores less than 105, we use the Z-score of 1 (as previously calculated for part b).
A Z-score of 1 corresponds to the lower 84.1% of the distribution.
- The percentage of scores less than 105 is 84.1%.
### h. Between 75 and 105
From part c, we know that the percentage of scores less than 75 is 15.9%.
From part f, we know that the percentage of scores less than 105 is 84.1%.
[tex]\[ \text{Percentage between 75 and 105} = 84.1\% - 15.9\% = 68.3\% \][/tex]
- The percentage of scores between 75 and 105 is 68.3%.
### Final Summary
a. The percentage of scores greater than 90 is 50.0%.
c. The percentage of scores less than 75 is 15.9%.
e. The percentage of scores less than 60 is 2.3%.
g. The percentage of scores greater than 75 is 84.1%.
b. The percentage of scores greater than 105 is 15.9%.
d. The percentage of scores less than 120 is 97.7%.
f. The percentage of scores less than 105 is 84.1%.
h. The percentage of scores between 75 and 105 is 68.3%.
