Here is the step-by-step solution to the problems based on the Normal Distribution rules provided in your worksheet.
Given Information:
*
Mean ($\mu$): 80
*
Standard Deviation ($\sigma$): 5
*
Total Students: 2000
First, let's map out the values on the bell curve using the standard deviations:
* Mean = 80
* +1 SD = $80 + 5 = 85$
* -1 SD = $80 - 5 = 75$
* +2 SD = $80 + 10 = 90$
* -2 SD = $80 - 10 = 70$
* +3 SD = $80 + 15 = 95$
* -3 SD = $80 - 15 = 65$
The Empirical Rule percentages:
* Between Mean and $\pm1$ SD:
34% (half of 68%)
* Between Mean and $\pm2$ SD:
47.5% (half of 95%)
* Between Mean and $\pm3$ SD:
49.85% (half of 99.7%)
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Step-by-Step Solutions
1. What percentage of the scores fall between 75 and 85?
* 75 is one standard deviation below the mean (-1 SD).
* 85 is one standard deviation above the mean (+1 SD).
* According to the Empirical Rule, the area within 1 standard deviation of the mean is
68%.
2. What percentage of the scores fall between 70 and 90?
* 70 is two standard deviations below the mean (-2 SD).
* 90 is two standard deviations above the mean (+2 SD).
* According to the Empirical Rule, the area within 2 standard deviations of the mean is
95%.
3. What percentage of the scores are greater than 80?
* 80 is the mean (the exact center of the curve).
* In a normal distribution, exactly half of the data lies above the mean and half lies below.
* Therefore, the percentage is
50%.
4. What percentage of the scores are between 70 and 85?
* We need to add two sections together: from 70 to 80, and from 80 to 85.
* From 80 to 85 (+1 SD): This is half of the 68% rule, which is
34%.
* From 70 to 80 (-2 SD to Mean): This is half of the 95% rule, which is
47.5%.
* Total: $34\% + 47.5\% =$
81.5%.
5. Approximately how many students scored higher than 85?
* First, find the percentage. The total area under the right side of the curve (scores > 80) is 50%.
* The area between 80 and 85 is 34%.
* So, the area greater than 85 is $50\% - 34\% =$
16%.
* Now, calculate the number of students: $16\%$ of 2000.
* Calculation: $0.16 \times 2000 = 320$.
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Final Answer:
1. 68%
2. 95%
3. 50%
4. 81.5%
5. 320 students
Parent Tip: Review the logic above to help your child master the concept of normal distribution worksheet with answers.
