Question

The frequency distribution shows the total number of points scored in the Super Bowl games from 1967 to 2012. Construct a histogram, frequency polygon, and ogive for the distribution. If you were to predict the total number of points for the next Super Bowl game, what range of values would you choose? In how many games were the total points greater than 56?

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
21-29	20.5-29.5	25	5
30-38	29.5-38.5	34	10
39-47	38.5-47.5	43	13
48-56	47.5-56.5	52	10
57-65	56.5-65.5	61	4
66-74	65.5-74.5	70	3
75-83	74.5-83.5	79	1
Total			46

 

Use the steps and draw the histogram of number of points scored in the Super Bowl game.

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

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 points scored in the Super Bowl game 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 points scored in the Super Bowl game.

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

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 points scored in the Super Bowl game is shown below:

 

The cumulative frequency is tabulated below:

	Cumulative Frequency
Less than 20.5	0
Less than 29.5	5
Less than 38.5	15
Less than 47.5	28
Less than 56.5	38
Less than 65.5	42
Less than 74.5	45
Less than 83.5	46

 

Use the steps and draw the Ogive of number of points scored in the Super Bowl game.

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

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 points scored in the Super Bowl game is shown below:

 

Find the number of games such that the total points were greater than 56.

 

The frequency for the interval 57-65 is 4, the frequency for the interval 66-74 is 3, and the frequency for the interval 75-83 is 1.

 

Hence, the number of games such that the total points were greater than 56 is given below:

4 + 3 + 1 = 8

 

Thus, 8 games have the total points greater than 56.

Thus, 8 games have the total points greater than 56.