# Using Data to Solve Problems

Warmup

Activity #1

Exploring One Variable Data graphing. • Study the data and answer the questions that follow.

Over a two-week period, Mai recorded the number of math homework problems she had each school day. Here are the results below: Calculate the following; Show your working.

(1.) The mean number of math homework problems.

(2.) The mean absolute deviation (MAD).

(3.) Interpret the mean and MAD. What do they tell you about the number of homework problems Mai had over these two weeks?

(4.) The median, quartile, maximum, and minimum of the same data on Mai’s math homework problems.

(5.) Which pair of measures of center and variability—mean and MAD, or median and IQR—do you think summarize the distribution of Mai’s math homework assignments better? Explain your reasoning.

You may use the applet below to help if you choose to. Enter the values needed to calculate the IQR and the mean when prompted.

Activity #2

`Represent Data on a Graph.`

Jada wanted to know whether a dot plot, a histogram, or a box plot would best summarize the center, variability, and other aspects of her homework data. • Use the axis below to make a dot plot to represent the data.
• Mark the position of the mean, which you calculated earlier, on the dot plot using a triangle (Δ).
• From the triangle, draw a horizontal line segment to the left and right sides to represent the MAD.
• Use the five-number summary from the previous task and the grid to draw a box plot that represents Jada’s homework data.

• Draw three histograms to represent Jada’s homework data. The width of the bars in each histogram should represent a different number of homework problems, as specified. The width of one bar represents 10 problems.

You can use the applet to make each type of graph if you choose to.

The width of one bar represents 5 problems.

The width of one bar represents 2 problems.

Which of the five representations should Jada use to summarize her data? Should she use a dot plot, box plot, or one of the histograms? Explain your reasoning.

Activity #3

`Construct a Histogram.`

Scientists studying the yellow perch, a species of fish, believe that the length of a fish is related to its age. This means that the longer the fish, the older it is. Adult yellow perch vary in size, but they are usually between 10 and 25 centimeters.

Scientists at the Great Lakes Water Institute caught, measured, and released yellow perch at several locations in Lake Michigan. The following summary is based on a sample of yellow perch from one of these locations. • Use the data to make a histogram that shows the lengths of the captured yellow perch. Each bar should contain the lengths shown in each row in the table.

(1.) How many fish were measured?

(2.) How do you know?

• Use the histogram to answer the following questions;

(3.) How would you describe the shape of the distribution?

(4.) Estimate the median length for this sample. Describe how you made this estimate.

(5.) Predict whether the mean length of this sample is greater than, less than, or nearly equal to the median length for this sample of fish? Explain your prediction.

(6.) Would you use the mean or the median to describe a typical length of the fish being studied? Explain your reasoning.

(7.) Based on your work so far, would you describe a typical age for the yellow perch in this sample as: “young,” “adult,” or “old”? Explain your reasoning.

(8.) Some researchers are concerned about the survival of the yellow perch. Do you think the lengths (or the ages) of the fish in this sample are something to worry about? Explain your reasoning.

Challenge #1

In one study on wild bears, researchers measured the head lengths and head widths, in inches, of 143 wild bears. The ages of the bears ranged from newborns (0 years) to 15 years. The box plots summarize the data from the study.