Percentages and Frequency of Scores
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Assume that a set of test scores is normally distributed with a mean of 100 and a standard deviation of 20. Use the 68-95-99.7 rule to find the following quantities:
Suggestion: make a drawing and label first
a. Percentage of scores less than 100
b. Relative frequency of scores less than 120
c. Percentage of scores less than 140
d. Percentage of scores less than 80
e. Relative frequency of scores less than 60
f. Percentage of scores greater than 120
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Solution Summary
The percentages and frequency of scores are examined.
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This is a problem related to the normal distribution which can be determined if the mean mu and the standard deviation sigma are given. The so-called 68-95-99.7 rule is the following. To make it clear, I denote mu=a and sigma=b. So,
P(a-b<X<a+b)=0.68 .........................(1)
P(a-2b<X<a+2b)=0.95 .......................(2)
P(a-3b<X<a+3b)=0.997 ......................(3)
As the normal curve is symmetric with respect to x=mu=a,
P(X<a)=P(X>a)=0.50 ...
Education
- BSc , Wuhan Univ. China
- MA, Shandong Univ.
Recent Feedback
- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"
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