Math Problem Statement
The physical plant at the main campus of a large state university receives daily requests to replace florescent light bulbs. the distribution of the number of daily requests is bell shaped and has a mean of 55 and a standard deviation of 8 using the empirical rule 68-95-99.7 what is the approximate percentage of the lightbulb replacements requests numbering between 47 and 55
Solution
The empirical rule (also known as the 68-95-99.7 rule) states that for a normal distribution:
- About 68% of the data falls within one standard deviation of the mean.
- About 95% falls within two standard deviations.
- About 99.7% falls within three standard deviations.
Given:
- Mean (μ) = 55
- Standard deviation (σ) = 8
We need to find the approximate percentage of requests between 47 and 55. First, let's express the range 47 to 55 in terms of standard deviations from the mean:
So, the interval from 47 to 55 covers the range from one standard deviation below the mean to the mean itself. Since the empirical rule tells us that 68% of the data falls within one standard deviation (from 47 to 63, in this case), half of that (i.e., from 47 to 55) will cover half of this 68% interval.
Thus, the approximate percentage of daily requests between 47 and 55 is:
Summary:
Approximately 34% of lightbulb replacement requests fall between 47 and 55.
Do you want more details or have any questions?
Here are 5 related questions to explore:
- What percentage of requests fall between 55 and 63?
- What is the percentage of requests falling between 47 and 63?
- How would the result change if the standard deviation was 10 instead of 8?
- What percentage of daily requests are below 47?
- What percentage of daily requests are above 63?
Tip: The empirical rule can be quickly applied to estimate probabilities in normally distributed data by using standard deviations.
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Math Problem Analysis
Mathematical Concepts
Normal Distribution
Empirical Rule (68-95-99.7 Rule)
Statistics
Formulas
Empirical rule percentages for normal distribution
Standard deviation (σ) from mean (μ)
Theorems
Empirical Rule (68-95-99.7 rule)
Suitable Grade Level
Grades 9-12
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