Math Problem Statement
The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped with a mean of 44 ounces and a standard deviation of 7 ounces. Using the Empirical Rule, answer the following questions:
a) 95% of the widget weights lie between __ and __.
b) What percentage of the widget weights lie between 23 and 58 ounces?
c) What percentage of the widget weights lie below 51?
Solution
Let's break down the problem based on the image you provided:
The Acme Company manufactures widgets with weights that are normally distributed.
- Mean (µ) = 44 ounces
- Standard deviation (σ) = 7 ounces
We will use the Empirical Rule (also known as the 68-95-99.7 rule) to answer the questions.
a) 95% of the widget weights lie between:
According to the Empirical Rule:
- 95% of data falls within 2 standard deviations of the mean.
Thus, the weights would lie between:
Calculating these values: [ 44 + 2(7) = 44 + 14 =
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Math Problem Analysis
Mathematical Concepts
Statistics
Normal Distribution
Empirical Rule
Formulas
Empirical Rule: For a normal distribution, approximately 68% of the data lies within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
Range for 95%: Mean ± 2 * Standard Deviation
Theorems
Empirical Rule
Suitable Grade Level
Grades 9-12