Math Problem Statement
In a mid size company in Boston the distribution of the number of phone calls answered each day by each of the 12 receptionist 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 percentage of daily phonecalls numbering between 47 and 63
Solution
The empirical rule, also known as the 68-95-99.7 rule, describes how data is distributed in a normal (bell-shaped) distribution:
- 68% of the data falls within 1 standard deviation of the mean.
- 95% of the data falls within 2 standard deviations of the mean.
- 99.7% of the data falls within 3 standard deviations of the mean.
Given the information:
- Mean (μ) = 55
- Standard Deviation (σ) = 8
Step 1: Find the range for 1 standard deviation
The empirical rule states that 68% of the data falls within 1 standard deviation of the mean. This means:
- Lower bound:
- Upper bound:
Step 2: Determine the percentage of phone calls between 47 and 63
Since 47 and 63 are exactly 1 standard deviation below and above the mean, the percentage of daily phone calls numbering between these two values is 68%.
Final Answer:
The percentage of daily phone calls answered between 47 and 63 is 68%.
Would you like more details on the empirical rule or its application? Here are some related questions that could expand your understanding:
- What is the percentage of phone calls answered between 39 and 71?
- How many receptionists are expected to answer between 47 and 63 calls daily?
- What percentage of receptionists answer more than 63 calls daily?
- What is the probability of a receptionist answering fewer than 47 calls?
- How does the empirical rule change if the standard deviation is different?
Tip: The empirical rule is very useful for quick estimates in normally distributed data without needing detailed calculations or statistical software.
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Math Problem Analysis
Mathematical Concepts
Statistics
Normal Distribution
Empirical Rule
Formulas
Empirical Rule: 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3 standard deviations
Standard deviation calculation: Upper bound = mean + 1 * standard deviation, Lower bound = mean - 1 * standard deviation
Theorems
Empirical Rule
Suitable Grade Level
Grades 10-12
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