Math Problem Statement

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 50 ounces and a standard deviation of 10 ounces. Use the Standard Deviation Rule (Empirical Rule). 99.7% of the widget weights lie between ______ and ______. What percentage of the widget weights lie between 40 and 80 ounces? What percentage of the widget weights lie below 70?

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 = μ3σ=503×10=5030=20\mu - 3\sigma = 50 - 3 \times 10 = 50 - 30 = 20 ounces.
  • Upper bound = μ+3σ=50+3×10=50+30=80\mu + 3\sigma = 50 + 3 \times 10 = 50 + 30 = 80 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:

  1. What percentage of widget weights lie between 30 and 70 ounces?
  2. What percentage of the weights lie above 60 ounces?
  3. How many standard deviations from the mean do 95% of the weights lie within?
  4. How would the range change if the standard deviation was 5 ounces instead of 10?
  5. 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!

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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)