Math Problem Statement
Observations are drawn from a bell-shaped distribution with a mean of 40 and a standard deviation of 5.
Approximately what percentage of the observations fall between 35 and 45?
Note: Round your answer to the nearest whole percent.
Approximately what percentage of the observations fall between 30 and 50?
Note: Round your answer to the nearest whole percent.
Approximately what percentage of the observations are less than 35?
Note: Round your answer to 1 decimal place.
Solution
To answer these questions, we'll apply the 68-95-99.7 rule (empirical rule) for normal distributions. This rule states that:
- 68% of the data falls within 1 standard deviation of the mean.
- 95% of the data falls within 2 standard deviations of the mean.
- 99.7% of the data falls within 3 standard deviations of the mean.
Given Information:
- Mean () = 40
- Standard Deviation () = 5
1. Percentage of observations between 35 and 45:
The range from 35 to 45 corresponds to 1 standard deviation below and above the mean.
- 35 is (1 standard deviation below the mean).
- 45 is (1 standard deviation above the mean).
Using the empirical rule, 68% of the observations lie between 35 and 45.
So, the percentage of observations between 35 and 45 is:
- 68%
2. Percentage of observations between 30 and 50:
The range from 30 to 50 corresponds to 2 standard deviations below and above the mean.
- 30 is (2 standard deviations below the mean).
- 50 is (2 standard deviations above the mean).
Using the empirical rule, 95% of the observations lie between 30 and 50.
So, the percentage of observations between 30 and 50 is:
- 95%
3. Percentage of observations less than 35:
The value 35 is 1 standard deviation below the mean.
- Since 68% of the data is within 1 standard deviation, we know that half of that (34%) is below the mean (40).
- Therefore, the percentage of observations less than 35 would be half of 68% (34%), plus the additional 34% below the mean, which makes it:
- 34% (below the mean) + 34% (half of the 68%) = 34%
Thus, the percentage of observations less than 35 is approximately:
- 34.0%
Summary of answers:
- 68% of the observations fall between 35 and 45.
- 95% of the observations fall between 30 and 50.
- 34.0% of the observations are less than 35.
Would you like further clarification or a breakdown of any of these calculations? Also, here are some follow-up questions:
- What is the percentage of observations between 40 and 45?
- How would you calculate the percentage of observations above 45?
- How do we calculate the percentage of observations between 30 and 40?
- What are the advantages of using the empirical rule for normal distributions?
- What happens if the distribution is not normal—can the empirical rule still be applied?
Tip: Remember that the empirical rule is most accurate for symmetric, bell-shaped (normal) distributions.
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Math Problem Analysis
Mathematical Concepts
Normal Distribution
Empirical Rule (68-95-99.7 rule)
Statistics
Formulas
Empirical Rule: 68% within 1 standard deviation, 95% within 2 standard deviations, 99.7% within 3 standard deviations
Theorems
Empirical Rule for Normal Distributions
Suitable Grade Level
Grades 9-12
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