Math Problem Statement
Solution
The problem presented uses the Empirical Rule (or 68-95-99.7 rule) for a normal distribution. The widget weights are normally distributed with a mean of 43 ounces and a standard deviation of 10 ounces.
Empirical Rule Breakdown:
- 68% of the data lies within 1 standard deviation (mean ± 1σ)
- 95% lies within 2 standard deviations (mean ± 2σ)
- 99.7% lies within 3 standard deviations (mean ± 3σ)
Part a) "95% of the widget weights lie between ___ and ___"
To find the range for 95% of the dataThe problem presented uses the Empirical Rule (or 68-95-99.7 rule) for a normal distribution. The widget weights are normally distributed with a mean of 43 ounces and a standard deviation of 10 ounces.
Empirical Rule Breakdown:
- 68% of the data lies within 1 standard deviation (mean ± 1σ)
- 95% lies within 2 standard deviations (mean ± 2σ)
- 99.7% lies within 3 standard deviations (mean ± 3σ)
Part a) "95% of the widget weights lie between ___ and ___"
To find the range for 95% of the
- Use 2 standard deviations from the mean:
- Lower bound = Mean - 2σ = 43 - 2(10) = 43 - 20 = 23 ounces
- Upper bound = Mean + 2σ = 43 + 2(10) = 43 + 20 = 63 ounces
Thus, 95% of the widget weights lie between 23 ounces and 63 ounces.
Part b) "What percentage of the widget weights lie between 13 and 63 ounces?"
To answer this:
- Lower bound: 13 is 3 standard deviations below the mean (43 - 3(10) = 13).
- Upper bound: 63 is 2 standard deviations above the mean (calculated above).
From the Empirical Rule:
- From -3σ to +2σ contains:
- 99.7% of the data lies between -3σ and +3σ.
- Since we're considering only up to +2σ, we subtract the last 0.15% of the tail beyond +2σ.
- So, the percentage of data between 13 and 63 ounces is 97.5%.
Final answers:
- a) 95% of the widget weights lie between 23 and 63 ounces.
- b) 97.5% of the widget weights lie between 13 and 63 ounces.
Would you like more details or have any questions about these calculations?
Here are 5 additional questions to deepen understanding:
- How would the percentages change if the standard deviation increased to 15 ounces?
- What does the Empirical Rule tell us about data points beyond 3 standard deviations from the mean?
- Can the normal distribution have negative values in this context? Why or why not?
- How would the range of weights differ if the percentage was 68% instead of 95%?
- How do we calculate probabilities for non-standard deviations?
Tip: The Empirical Rule is a great approximation for normal distributions, but for more precision, especially for non-integer standard deviations, using the Z-score tables is recommended.
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Math Problem Analysis
Mathematical Concepts
Normal Distribution
Empirical Rule
Standard Deviation
Formulas
Lower bound = Mean - 2σ
Upper bound = Mean + 2σ
Theorems
Empirical Rule
Suitable Grade Level
Grades 9-12
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