Question

The number of bears killed in 2010 for 52 counties in Pennsylvania is shown in the frequency distribution. Construct a histogram, frequency polygon, and ogive for the data. Comment on the skewness of the distribution. How many counties had 75 or fewer bears killed?

Answer 1

Compute the class boundaries, and midpoints (Xm) for the given class limits.

 

The general formula for the class boundaries and midpoint are given below

Lower Boundary = Lower Limit – 0.5

Upper Boundary = Upper Limit + 0.5

Xm = (Lower limit + Upper limit)/2

 

The class boundaries and midpoints (Xm) are tabulated below:

Class limits	Boundaries	Midpoint (Xm)	Frequency
1-25	0.5-25.5	13	18
26-50	25.5-50.5	38	12
51-75	50.5-75.5	63	7
76-100	75.5-100.5	88	6
101-125	100.5-125.5	113	2
126-150	125.5-150.5	138	3
151-175	150.5-175.5	163	0
176-200	175.5-200.5	188	2
201-225	200.5-225.5	213	0
226-250	225.5-250.5	238	2
Total			52

 

Use the steps and draw the histogram of number of bears killed in 2010.

Step 1: Draw the horizontal line (x axis) and label it as Number of bears.

Step 2: Represent the class boundaries on the x axis.

Step 3: Draw the vertical line (y axis) and label it as Frequency.

Step 4: Using the frequencies as the heights, draw the vertical bars for each class.

 

The histogram of number of bears killed in 2010 is shown below:

 

Observation:

From the above histogram, it is clear that the right side of the histogram is extended larger than the left side. Hence, the distribution seems to be positively skewed.

 

Use the steps and draw the frequency polygon of number of bears killed in 2010.

Step 1: Draw the horizontal line (x axis) and label it as Number of bears.

Step 2: Represent the midpoint of each class on the x axis.

Step 3: Draw the vertical line (y axis) and label it as Frequency.

Step 4: Plot the points using midpoints and frequencies.

Step 5: Join the adjacent points with line segments.

Step 6: Extend the line at the beginning and end of the graph where the earlier and next midpoints would be located on the x axis.

 

The frequency polygon of number of bears killed in 2010 is shown below:

 

The cumulative frequency is tabulated below:

	Cumulative Frequency
Less than 0.5	0
Less than 25.5	18
Less than 50.5	30
Less than 75.5	37
Less than 100.5	43
Less than 125.5	45
Less than 150.5	48
Less than 175.5	48
Less than 200.5	50
Less than 225.5	50
Less than 250.5	52

 

Use the steps and draw the Ogive of number of bears killed in 2010.

Step 1: Draw the horizontal line (x axis) and label it as Number of bears.

Step 2: Represent the upper class boundaries on the x axis.

Step 3: Draw the vertical line (y axis) and label it as Cumulative frequency.

Step 4: Plot the points using upper class boundaries and cumulative frequencies.

Step 5: Join the adjacent points with line segments.

Step 6: Extend the line to the first lower class boundary on the x axis.

 

The Ogive of number of bears killed in 2010 is shown below:

 

Find the number of counties which have 75 or fewer bears killed.

The frequency for the interval 1-25 is 18, the frequency for the interval 26-50 is 12, and the frequency for the interval 50-75 is 7.

 

Hence, the number of counties which have 75 or fewer bears killed is given below:

18 + 12 + 7 = 37

 

Thus, 37 counties had 75 or fewer bears killed.

Thus, 37 counties had 75 or fewer bears killed.