The accompanying data represent the homework scores for material on Polynomial and Rational Functions for a random sample of students in a college algebra course. Complete parts (a) through (i).
Click the icon to view the data.
(3) Draw a relative frequency histogram of the data. Choose the correct answer below.
A. B.
Homomots Scores
c.
Homework Scores
(c) Determine the mean and median score.
The mean enrore is $\square$ . and the median score it $\square$ .
(Type integers or decimals. Do not round.)
Answer (2)
To find the mean and median of the homework scores, calculate the midpoints for each interval, then determine the mean by using the relative frequencies of these midpoints. For the median, calculate cumulative relative frequencies to identify the median interval and use the median formula for grouped data. The mean and median scores will provide insights into the students' performance.
;
Calculate the midpoints of each interval.
Compute the mean by multiplying each midpoint by its relative frequency and summing the results: m e an = 73.14 .
Determine the cumulative relative frequencies to identify the median interval.
Apply the formula for the median of grouped data to find the median score: m e d ian = 81.2 .
Explanation
Understanding the Problem The problem provides a frequency distribution of homework scores and asks us to calculate the mean and median score. The data is grouped into intervals, so we'll need to use the grouped data formulas to estimate these statistics.
Calculating Midpoints First, we need to find the midpoint of each interval. The midpoints are calculated as follows:
50-59: 2 50 + 59 = 54.5
60-69: 2 60 + 69 = 64.5
70-79: 2 70 + 79 = 74.5
80-89: 2 80 + 89 = 84.5
90-99: 2 90 + 99 = 94.5
These midpoints represent the average score within each interval.
Calculating the Mean Next, we calculate the mean score by multiplying each midpoint by its relative frequency and summing the results:
m e an = ( 54.5 × 0.02 ) + ( 64.5 × 0.15 ) + ( 74.5 × 0.3 ) + ( 84.5 × 0.25 ) + ( 94.5 × 0.2 ) m e an = 1.09 + 9.675 + 22.35 + 21.125 + 18.9 m e an = 73.14
So, the estimated mean score is 73.14.
Finding the Median Interval To find the median, we first need to determine the cumulative relative frequencies:
50-59: 0.02
60-69: 0.02 + 0.15 = 0.17
70-79: 0.17 + 0.3 = 0.47
80-89: 0.47 + 0.25 = 0.72
90-99: 0.72 + 0.2 = 0.92
The median is the score that divides the distribution into two equal halves. Since the cumulative relative frequency reaches 0.5 in the 80-89 interval, this is the median interval.
Calculating the Median Now we use the formula for the median of grouped data:
m e d ian = L + f 0.5 − c f × w
Where:
L is the lower limit of the median interval (80)
cf is the cumulative relative frequency of the interval before the median interval (0.47)
f is the relative frequency of the median interval (0.25)
w is the width of the interval (10)
m e d ian = 80 + 0.25 0.5 − 0.47 × 10 m e d ian = 80 + 0.25 0.03 × 10 m e d ian = 80 + 0.12 × 10 m e d ian = 80 + 1.2 m e d ian = 81.2
Therefore, the estimated median score is 81.2.
Final Answer The mean score is 73.14, and the median score is 81.2.
Examples
Understanding the distribution of scores, like homework grades, can help teachers tailor their instruction. For example, if the median score is higher than the mean, it suggests that there are some lower scores pulling the average down, and the teacher might focus on providing extra support to those students. This kind of analysis is also used in business to understand customer satisfaction, in healthcare to analyze patient outcomes, and in many other fields where data is grouped and analyzed to make informed decisions. By calculating measures like mean and median, we gain valuable insights into the overall performance and identify areas for improvement.
