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# Finding lower and upper limits based on a known distribution

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A nationwide test taken by high school sophomores and juniors has three sections, each scored on a scale of 20 to 80. In a recent year, the national mean score for the writing section was 49.6, with a standard deviation of 10.9. Based on this information, complete the following statements about the distribution of the scores on the writing section for the recent year.

1. According to Chebyshev's theorem, at least what percent of the scores lie between 27.8 and 71.4?
2. According to Chebyshev's theorem, at least 89% of the scores lie between _ and _ (round your answer to 1 decimal place.)
3. Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the scores lie between _ and _
4. Suppose that the distribution is bell-shaped. According to the empirical rule approximately what percent of the scores lie between 27.8 and 71.4?

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The solution contains detailed explanation of finding lower and upper limits based on a known distribution.

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• 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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