The organizers of the Red Maple College Fun Run gathered data about the runners' ages. This box plot shows the results.
Runners' ages
15
25
35
45
55
65
What percent of the runners were between 20 and 45 years old?
To answer this question, we need to determine the interquartile range (IQR) of the data, which represents the middle 50% of the ages. From the box plot, we can see that the lower quartile (Q1) is 25 and the upper quartile (Q3) is 45. Therefore, the IQR is 45 - 25 = 20.
Next, we can find the proportion of runners whose ages fall between 20 and 45 by dividing the length of this interval by the length of the entire IQR:
Proportion = (45 - 20) / 20 = 1.25
This means that the interval from 20 to 45 years old is 1.25 times as long as the middle 50% of the ages. To convert this proportion to a percentage, we can multiply by 100:
Percent = Proportion x 100% = 1.25 x 100% = 125%
However, this result doesn't make sense because a percentage should be between 0% and 100%. The reason for the incorrect result is that the interval from 20 to 45 includes some ages outside the IQR, specifically 15 and 55. To correct for this, we need to determine the proportion of the entire data set that falls in the interval from 20 to 45.
From the box plot, we can see that the minimum age is 15 and the maximum age is 65. Therefore, the range of the data is 65 - 15 = 50. The interval from 20 to 45 represents a segment of this range that is 25 units long. To find the percentage of runners whose ages fall in this segment, we can divide its length by the total range of the data:
Percent = (45 - 20) / (65 - 15) x 100% = 25 / 50 x 100% = 50%
Therefore, 50% of the runners were between 20 and 45 years old.
Last year, Nicholas started babysitting to earn extra spending money. Over the summer, he tracked how much money he earned at each babysitting job. This box plot shows the results.
Babysitting earnings ($)
10
20
30
40
50
60
What percent of the time did Nicholas earn $25 or less?
To answer this question, we need to use the box plot to determine the quartiles of the data. The lower quartile (Q1) is the point below which 25% of the earnings lie, and the upper quartile (Q3) is the point below which 75% of the earnings lie. From the box plot, we can see that:
- Q1 = $20
- Q3 = $50
Therefore, the interquartile range (IQR) is $50 - $20 = $30.
To determine the percentage of the time that Nicholas earned $25 or less, we need to determine what proportion of the data falls within that range. Since $25 is between $20 and $30, it lies within the IQR. We know that the middle 50% of the data falls within the IQR, so we need to determine what proportion of the IQR falls between $20 and $25.
The distance from $20 to $25 is $5, which is one-sixth of the IQR. Therefore, one-sixth of the middle 50% of earnings falls between $20 and $25. This means that the proportion of the data that falls in this range is:
Proportion = 1/6 x 1/2 = 1/12
To convert this proportion to a percentage, we multiply by 100:
Percent = Proportion x 100% = 1/12 x 100% = 8.33%
Therefore, Nicholas earned $25 or less about 8.33% of the time.
The coach of Darnel's baseball team recorded how many runs the team scored in each game last season. This box plot shows the results.
Runs scored
0
2
4
6
8
10
In what fraction of their games did Darnel's team score 2 or more runs?
To answer this question, we need to determine the quartiles of the data from the box plot. The lower quartile (Q1) is the point below which 25% of the data lie. From the box plot, we can see that Q1 is 2 runs.
Therefore, in 25% of the team's games, they scored 2 runs or less. We want to determine the fraction of games in which the team scored 2 or more runs, which is equivalent to finding the complement of this probability (i.e. the probability that they scored less than 2 runs and subtracting it from 1).
The probability that the team scored less than 2 runs is 25%, so the probability that they scored 2 or more runs is:
Probability = 1 - 0.25 = 0.75
Therefore, the fraction of Darnel's team games in which they scored 2 or more runs is 0.75, or 3/4.
Last year, Janelle worked as a party planner. She kept track of the number of guests at each of the parties she planned. This box plot shows the results.
Party guests
0
100
200
300
400
500
What fraction of the parties had 100 or more guests?
To answer this question, we need to determine the quartiles of the data from the box plot. The lower quartile (Q1) is the point below which 25% of the data lie. From the box plot, we can see that Q1 is 100 guests.
Therefore, in 25% of the parties Janelle planned, they had 100 guests or fewer. We want to determine the fraction of parties in which there were 100 or more guests, which is equivalent to finding the complement of this probability (i.e. the probability that there were fewer than 100 guests and subtracting it from 1).
The probability that there were fewer than 100 guests is 25%, so the probability that there were 100 or more guests is:
Probability = 1 - 0.25 = 0.75
Therefore, the fraction of parties that had 100 or more guests is 0.75, or 3/4.