1. A 5-number summary for number of hours worked by employees of a company in a week were: (pts) Min 12 Q1 23 Median 40 Q3 53 Max 60 a) What is the value such that 25% of the emplo... 1. A 5-number summary for number of hours worked by employees of a company in a week were: (pts) Min 12 Q1 23 Median 40 Q3 53 Max 60 a) What is the value such that 25% of the employees worked more than that value? b) Calculate the range and interquartile range (IQR) of number of hours worked by the employees. c) About what percent of the employees worked between 23 and 53 hours? Give a complete sentence.
Understand the Problem
The question is asking for three specific statistical calculations based on a five-number summary of hours worked by employees: the value such that 25% of the employees worked more, the range and interquartile range of the hours worked, and the percentage of employees who worked between two specific hours.
Answer
a) $23$ \newline b) Range: $48$, IQR: $30$ \newline c) $50\%$
Answer for screen readers
a) The value such that 25% of the employees worked more than is $23$.
b) The range is $48$, and the interquartile range (IQR) is $30$.
c) About $50%$ of the employees worked between $23$ and $53$ hours.
Steps to Solve
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Determine the value for 25% of employees working more From the five-number summary, the first quartile (Q1) represents the value where 25% of the data falls below it. Therefore, the value such that 25% of employees worked more is: $$ Q1 = 23 $$
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Calculate the range The range of a dataset is calculated as the difference between the maximum and minimum values: $$ \text{Range} = \text{Max} - \text{Min} $$ Using the values from the summary: $$ \text{Range} = 60 - 12 = 48 $$
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Calculate the interquartile range (IQR) The IQR is the difference between the third quartile (Q3) and the first quartile (Q1): $$ \text{IQR} = Q3 - Q1 $$ Using the values from the summary: $$ \text{IQR} = 53 - 23 = 30 $$
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Calculate the percentage of employees working between 23 and 53 hours From the five-number summary:
- Q1 = 23 corresponds to 25% of employees
- Q3 = 53 corresponds to 75% of employees
To find the percentage of employees working between 23 and 53 hours: $$ \text{Percentage} = Q3 - Q1 = 75% - 25% = 50% $$ Therefore, approximately 50% of the employees worked between 23 and 53 hours.
a) The value such that 25% of the employees worked more than is $23$.
b) The range is $48$, and the interquartile range (IQR) is $30$.
c) About $50%$ of the employees worked between $23$ and $53$ hours.
More Information
A five-number summary provides a quick overview of the distribution of data. Each number (min, Q1, median, Q3, max) gives critical insights into the spread and center of the dataset. Knowing how to calculate the range and IQR helps in understanding the variability of the data.
Tips
- Confusing Q1 with the median: Remember Q1 is the first quartile, not the median.
- Miscalculating the range: Ensure to subtract the minimum from the maximum accurately.
- Overlooking percentage calculations: Ensure you understand the positions of Q1 and Q3 when calculating the percentage of employees within ranges.
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