Math Problem Statement

During one shift, the express lane clerk recorded how many times customers violated the "10 items or less" rule for his lane. In particular, he recorded how many items over the limit each violator placed on the conveyor belt. This data is summarized in the histogram below. NOTE: The last class actually represents "7 or more items," not just 7 items.

2 4 6 8 10 12 14 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5

items over 10

Frequency [Graphs generated by this script: setBorder(54,40,20,5); initPicture(0,7.5,0,14);axes(1000,2,1,1000,2); fill="blue"; stroke="black";line([0.5,-0.28],[0.5,0.28]); text([0.5,0],"0.5","below");line([1.5,-0.28],[1.5,0.28]); text([1.5,0],"1.5","below");line([2.5,-0.28],[2.5,0.28]); text([2.5,0],"2.5","below");line([3.5,-0.28],[3.5,0.28]); text([3.5,0],"3.5","below");line([4.5,-0.28],[4.5,0.28]); text([4.5,0],"4.5","below");line([5.5,-0.28],[5.5,0.28]); text([5.5,0],"5.5","below");line([6.5,-0.28],[6.5,0.28]); text([6.5,0],"6.5","below");line([7.5,-0.28],[7.5,0.28]); text([7.5,0],"7.5","below");textabs([165,0],"# items over 10","above");textabs([0,115],"Frequency","right",90);rect([0.5,0],[1.5,6]);rect([1.5,0],[2.5,9]);rect([2.5,0],[3.5,11]);rect([3.5,0],[4.5,14]);rect([4.5,0],[5.5,12]);rect([5.5,0],[6.5,8]);rect([6.5,0],[7.5,5]);]

What is the frequency of times the limit was exceeded by less than 4 items?

Solution

The histogram represents the frequency of customers exceeding the "10 items or less" rule, divided by how many items they exceeded the limit by. The x-axis represents the number of items over the limit, and the y-axis shows the frequency.

We are asked to find the frequency of times the limit was exceeded by less than 4 items.

This includes the bars for:

  • 0.5 to 1.5 items over: frequency of 6
  • 1.5 to 2.5 items over: frequency of 9
  • 2.5 to 3.5 items over: frequency of 11

Now, adding these frequencies together:

6+9+11=266 + 9 + 11 = 26

Thus, the frequency of times the limit was exceeded by less than 4 items is 26.

Would you like to know more details or have any questions? Here are some related questions:

  1. How do we interpret histograms in general?
  2. What is the significance of the "7 or more items" class in this context?
  3. How do you calculate cumulative frequencies in a histogram?
  4. How could the average number of items over the limit be calculated from this histogram?
  5. What would be the effect of having a stricter or more lenient rule (e.g., changing to 12 items)?

Tip: When interpreting histograms, always note the labels and ranges on both axes to understand the frequency distribution properly.

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Math Problem Analysis

Mathematical Concepts

Histogram interpretation
Frequency distribution
Summation

Formulas

Sum of frequencies for specific intervals

Theorems

Basic histogram interpretation principles

Suitable Grade Level

Grades 6-8

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