Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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%.
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%.
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
