Question

Sixteen high school students completed a class to prepare them for the College Boards. Their scores were as follows. \(\begin{array}{llll}420 & 345 & 560 & 650 \\ 459 & 499 & 500 & 657 \\ 467 & 480 & 505 & 555 \\ 480 & 520 & 530 & 589\end{array}\) These same 16 students were given a test of math and verbal ability to measure their readiness for college-level work. Scores are reported here in terms of the percentage of correct answers for each test. \begin{tabular}{clcc} \hline \multicolumn{4}{c} { Math Test } \\ \hline 67 & 45 & 68 & 70 \\ 72 & 85 & 90 & 99 \\ 50 & 73 & 77 & 78 \\ 52 & 66 & 89 & 75 \\ \hline \multicolumn{4}{l} { Verbal Test } \\ \hline 89 & 90 & 78 & 77 \\ 75 & 70 & 56 & 60 \\ 77 & 78 & 80 & 92 \\ 98 & 72 & 77 & 82 \end{tabular} a. Display each of these variables in a frequency distribution with columns for percentages and cumulative percentages. b. Construct a histogram and frequency polygon for these data. c. (This problem is optional.) Find the upper and lower real limits for the intervals you established.

Short answer

1. Create frequency tables with intervals, frequencies, percentages, and cumulative percentages. 2. Draw histograms and frequency polygons. Optional: Calculate real limits for intervals.

Step-by-step solution

- Create Frequency Distribution Tables

Start by creating frequency distribution tables for the three sets of scores: College Board Scores, Math Test Scores, and Verbal Test Scores. Each table should include columns for the score intervals, frequencies, percentages, and cumulative percentages.

- Determine Score Intervals

Divide the range of scores for each set into intervals. A common practice is to use intervals of equal width. For example, for the College Board Scores ranging from 345 to 657, you might use intervals of 50 points: 340-389, 390-439, etc.

- Calculate Frequencies

Count the number of scores that fall into each interval. Record these frequencies in the table.

- Calculate Percentages

Calculate the percentage for each interval using the formula: \(\frac{\text{frequency}}{\text{total number of scores}} \times 100\). Record these percentages in the table.

- Calculate Cumulative Percentages

Calculate the cumulative percentages by adding the percentage of each interval to the sum of the percentages of all previous intervals. Record these cumulative percentages in the table.

- Construct Histograms

Draw histograms for each set of scores using the frequency data. Label the x-axis with score intervals and the y-axis with frequencies.

- Construct Frequency Polygons

For each histogram, construct a frequency polygon by plotting the midpoints of each interval at the corresponding frequency and connecting the points with straight lines. Add points at the beginning and end of the graph to close the polygon.

(Optional) - Find Real Limits

To find the upper and lower real limits for each interval, you can add and subtract half of the smallest measurement unit in the data. For example, if scores are given as whole numbers, use 0.5 as the smallest unit. For an interval 340-389, the lower real limit is 340 - 0.5 = 339.5, and the upper real limit is 389 + 0.5 = 389.5.

Key concepts

histogram construction

Creating a histogram is a great way to visually display data distribution.
Start by setting up your x-axis with the score intervals and y-axis with frequencies.
For example, using College Board Scores from the exercise, an interval might be 340-389, 390-439, etc.
First, count how many scores fall into each interval (this is the frequency).
Draw rectangular bars for each interval. The height of each bar should correspond to the frequency of scores in that interval.
Don't forget to label your axes.
Histograms make it easy to see which intervals are most common and to identify any patterns in the data.

cumulative percentage

Cumulative percentage helps understand the progressive accumulation of data.
To calculate it, start with your frequency table.
Remember, the percentage for each interval is calculated as follows: \(\frac{\text{frequency}}{\text{total number of scores}} \times 100\).
Add the percentage from the first interval to the percentage of the next interval, and continue doing this through the table.
For example, if your frequency table for Math Test Scores shows intervals with percentages of 10%, 20%, and 30%, then the cumulative percentages will be 10%, 30% (10%+20%), and 60% (30%+30%).
Cumulative percentages can show us not just how many scores fall into an interval but how many scores fall below a certain threshold.

real limits calculation

Real limits help provide accurate boundaries for intervals.
They are especially important when your data are continuous.
To find real limits, add and subtract half the measurement unit from the boundaries of each interval.
For instance, in the interval 340-389, scores are whole numbers, so the measurement unit is 1.
Half of this unit is 0.5.
The lower real limit is then 340 - 0.5 = 339.5, and the upper real limit is 389 + 0.5 = 389.5.
This ensures that every possible value in the interval, including boundary values, is considered.