Photo AI
The following table gives the distribution of donations, in euro, made by 20 people to an appeal fund: Amount of donation, € 5 - 15 15 - 25 25 - 35 35 - 65 Number of people 2 4 8 6 (Note: 5 - 15 means 5 or over but less than 15 €) (i) Draw a histogram to represent the data - Leaving Cert Mathematics - Question 7 - 2010
Question 7
The following table gives the distribution of donations, in euro, made by 20 people to an appeal fund: Amount of donation, € 5 - 15 15 - 25 25 - 35... show full transcript
Worked Solution & Example Answer:The following table gives the distribution of donations, in euro, made by 20 people to an appeal fund: Amount of donation, € 5 - 15 15 - 25 25 - 35 35 - 65 Number of people 2 4 8 6 (Note: 5 - 15 means 5 or over but less than 15 €) (i) Draw a histogram to represent the data - Leaving Cert Mathematics - Question 7 - 2010
Step 1
Draw a histogram to represent the data.
Answer
To draw a histogram for the data, we need to represent the mid-intervals on the x-axis and the frequency (number of people) on the y-axis. The mid-intervals are:
- 5 - 15 (€5 to €15): 2 people
- 15 - 25 (€15 to €25): 4 people
- 25 - 35 (€25 to €35): 8 people
- 35 - 65 (€35 to €65): 6 people
The histogram would consist of bars for each interval with heights corresponding to the frequencies. Each bar's width should represent the class interval, and the bars should not have spaces between them.
Step 2
Taking mid-interval values, calculate the mean amount donated.
Answer
To calculate the mean amount donated, we first find the mid-value for each class:
- For 5 - 15, mid-value = (5 + 15) / 2 = 10
- For 15 - 25, mid-value = (15 + 25) / 2 = 20
- For 25 - 35, mid-value = (25 + 35) / 2 = 30
- For 35 - 65, mid-value = (35 + 65) / 2 = 50
Now we calculate the total amount donated:
egin{align*} ext{Total Amount} &= (10 imes 2) + (20 imes 4) + (30 imes 8) + (50 imes 6) \ &= 20 + 80 + 240 + 300 \ &= 640 ext{ euros} ext{Total Number of People} &= 20 ext{Mean Amount Donated} &= rac{640}{20} = 32 ext{ euros} \end{align*}
Step 3
Taking mid-interval values, calculate the standard deviation, correct to one decimal place.
Answer
First, we find the deviations from the mean for each mid-interval:
- For 5 - 15: deviation = 10 - 32 = -22
- For 15 - 25: deviation = 20 - 32 = -12
- For 25 - 35: deviation = 30 - 32 = -2
- For 35 - 65: deviation = 50 - 32 = 18
Next, we square these deviations and multiply by the number of people:
egin{align*} ext{Variance} &= rac{ ext{Sum of squared deviations}}{N} \ &= rac{(2 imes (-22)^2) + (4 imes (-12)^2) + (8 imes (-2)^2) + (6 imes (18)^2)}{20} \ &= rac{(2 imes 484) + (4 imes 144) + (8 imes 4) + (6 imes 324)}{20} \ &= rac{968 + 576 + 32 + 1944}{20} \ &= rac{3520}{20} = 176 \ ext{Standard Deviation} &= ext{sqrt(Variance)} = ext{sqrt(176)} \ & ext{≈ 13.3} (to one decimal place) \end{align*}
Step 4
Draw the cumulative frequency curve.
Answer
To draw the cumulative frequency curve, plot the scores on the x-axis and the cumulative frequencies on the y-axis. Mark points for each cumulative frequency against the corresponding score:
- At ≤ 20, cumulative frequency = 40
- At ≤ 40, cumulative frequency = 150
- At ≤ 60, cumulative frequency = 300
- At ≤ 80, cumulative frequency = 380
- At ≤ 100, cumulative frequency = 400
After plotting these points, connect them smoothly to form the cumulative frequency curve.
Step 5
Use your curve to estimate the median score.
Answer
To estimate the median score from the cumulative frequency curve, locate the point corresponding to half the total number of candidates (which is 200) on the y-axis. The point on the curve where it intersects with 200 then leads to the corresponding x value, which represents the median score.
Step 6
Candidates who scored above 65 in the test were called for interview. Use your curve to estimate the number of candidates who were called for interview.
Answer
To estimate the number of candidates who scored above 65, first find the score of 65 on the x-axis of the cumulative frequency curve. Find the corresponding cumulative frequency for this score and subtract it from the total number of candidates (400). If the cumulative frequency at 65 is, for example, 325, then:
egin{align*} ext{Candidates above 65} &= 400 - 325 = 75 ext{ candidates} \end{align*}