Math Problem Statement
Solution
Let's go through the questions step by step using the Empirical Rule (also known as the 68-95-99.7 Rule) and the provided information:
- Mean (μ) = 50 ounces
- Standard deviation (σ) = 10 ounces
- The distribution is bell-shaped (normal distribution).
a) 99.7% of the widget weights lie between ______ and ______.
According to the Empirical Rule:
- 99.7% of the data falls within three standard deviations (±3σ) from the mean.
So, the range would be:
- Lower bound = ounces.
- Upper bound = ounces.
Thus, 99.7% of the widget weights lie between 20 and 80 ounces.
b) What percentage of the widget weights lie between 40 and 80 ounces?
- The range from 40 to 80 encompasses 40 (which is one standard deviation below the mean, μ - 1σ) to 80 (which is three standard deviations above the mean, μ + 3σ).
Using the Empirical Rule:
- 68% of the data lies within 1 standard deviation (between 40 and 60).
- 95% of the data lies within 2 standard deviations (between 30 and 70).
- 99.7% of the data lies within 3 standard deviations (between 20 and 80).
Since we're looking for the percentage between 40 and 80:
- From 40 to 50 (mean), it covers half of 68% (so, 34%).
- From 50 to 80, this covers the remaining part of 99.7%, which is 49.85% (half of 99.7%).
Thus, the total percentage from 40 to 80 is 34% + 49.85% = 83.85% (approximately 84%).
c) What percentage of the widget weights lie below 70?
- 70 is 2 standard deviations above the mean (μ + 2σ).
Using the Empirical Rule:
- 95% of the data lies within two standard deviations (between 30 and 70).
- Since the distribution is symmetrical, half of this data lies below 70, which is 95% of the total.
Thus, 97.5% of the widget weights lie below 70 ounces.
Would you like more details or have any questions?
5 Related Questions:
- What percentage of widget weights lie between 30 and 70 ounces?
- What percentage of the weights lie above 60 ounces?
- How many standard deviations from the mean do 95% of the weights lie within?
- How would the range change if the standard deviation was 5 ounces instead of 10?
- What percentage of widget weights lie below 30 ounces?
Tip:
The Empirical Rule is very useful for approximating probabilities in normal distributions without needing detailed calculations!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Normal Distribution
Empirical Rule (68-95-99.7 Rule)
Standard Deviation
Formulas
Lower Bound = Mean - 3 * Standard Deviation
Upper Bound = Mean + 3 * Standard Deviation
Theorems
Empirical Rule (68-95-99.7 Rule)
Suitable Grade Level
High School (Grades 9-12)
Related Recommendation
Empirical Rule and Normal Distribution Problem: Widget Weights
Normal Distribution Empirical Rule Problem: Widget Weights
Empirical Rule - Widget Weight Distribution with Mean of 46 Ounces and Standard Deviation of 10 Ounces
Empirical Rule and Widget Weights: Finding the 99.7% Range
Widget Weight Distribution: Empirical Rule & Standard Deviation