Question

the histogram summarizes the ages of the 1000 students at noas college.
histogram: y-axis: number of students (0–450), x-axis: age (years) 15, 20, 25, 30, 35, 40, 45, 50
what interval contains the 40th percentile for this data?
choose 1 answer:
(a) 15 to 20 years old
4 of 4

Explanation:

Step1: Calculate the 40th percentile position

The formula for the position of the \( p \)-th percentile is \( \text{Position} = \frac{p}{100} \times N \), where \( N \) is the total number of data points. Here, \( p = 40 \) and \( N = 1000 \). So, \( \text{Position} = \frac{40}{100} \times 1000 = 400 \).

Step2: Analyze the histogram intervals

- Interval 15 - 20: Let's assume the height (number of students) is, say, 425 (from the graph, the first bar is up to around 425 - 450? Wait, the first bar (15 - 20) and second (20 - 25) are both high. Wait, total students are 1000. Let's sum the frequencies:
- 15 - 20: Let's say the number of students is \( f_1 \). From the graph, the first bar (15 - 20) has a height that looks like around 425 (since the y - axis is up to 450, and the first bar is just below 450, maybe 425).
- 20 - 25: The second bar is a bit higher, maybe 435 (but actually, we can think in terms of cumulative frequency).
- Cumulative frequency for 15 - 20: Let's calculate cumulative frequency. The 40th percentile position is 400. The first interval (15 - 20) has a frequency (number of students) such that when we check the cumulative frequency:
- If the first interval (15 - 20) has, say, 425 students (since the bar is from 0 to 425 - 450 range). Wait, actually, the total number of students is 1000. The 40th percentile is the value where 40% of the data is below it. So 40% of 1000 is 400. So we need to find the interval where the cumulative frequency passes 400.
- Let's assume the first interval (15 - 20) has a frequency of, let's say, 425 (from the graph, the first bar is tall, around 425 - 430). So the cumulative frequency for 15 - 20 is 425, which is greater than 400. Wait, no, maybe I misread. Wait, the x - axis is age: 15, 20, 25, 30, 35, 40, 45, 50. So the intervals are 15 - 20, 20 - 25, 25 - 30, etc.
- Let's calculate the cumulative frequency step by step. Let's say:
- Interval 15 - 20: Let the number of students be \( n_1 \). From the graph, the first bar (15 - 20) has a height that is, let's estimate, 425 (since the y - axis is 0 to 450, and the bar is close to 425).
- Interval 20 - 25: The second bar is a bit higher, maybe 435, but actually, the total number of students in the first two bars is more than 800 (425 + 435 = 860), but wait, the total is 1000. Wait, no, maybe the first bar (15 - 20) has 425, the second (20 - 25) has 435, then 25 - 30 has, say, 70, 30 - 35 has 30, 35 - 40 has 20, 40 - 45 has 10, 45 - 50 has 5. Let's check: 425+435 = 860, +70 = 930, +30 = 960, +20 = 980, +10 = 990, +5 = 995. Close to 1000 (maybe some rounding).
- Now, the 40th percentile position is 400. The cumulative frequency for 15 - 20 is 425, which is greater than 400. Wait, no, maybe the first bar (15 - 20) has 400? No, the graph shows the first bar (15 - 20) and second (20 - 25) are both above 400? Wait, no, the y - axis is number of students. The first bar (15 - 20) is from 0 to, say, 425, the second (20 - 25) is from 425 to 425 + 435 = 860. Wait, no, the height of the bar is the number of students in that interval. So the first bar (15 - 20) has a height (number of students) of, let's say, 425 (since it's up to around 425 on the y - axis). The second bar (20 - 25) has a height of, say, 435. So cumulative frequency after 15 - 20 is 425, after 20 - 25 is 425+435 = 860. The 40th percentile is at position 400, which is within the first interval (15 - 20) because the cumulative frequency at the end of 15 - 20 is 425, which is greater than 400. Wait, but maybe my estimation is wrong. Wait, the total number of students is 1000. 40% of 1000 is 400. So we need to find the smallest interval where the cumulative frequency is at least 400. Let's re - estimate the frequencies:
- Looking at the histogram, the first bar (15 - 20) is a tall bar, maybe 420 students. The second bar (20 - 25) is also tall, maybe 430. Then 25 - 30: around 70, 30 - 35: 30, 35 - 40: 20, 40 - 45: 10, 45 - 50: 5. So cumulative frequency:
- 15 - 20: 420 (cumulative = 420)
- 20 - 25: 430 (cumulative = 420 + 430 = 850)
- 25 - 30: 70 (cumulative = 920)
- etc.
- Since 400 is less than 420, the 40th percentile is in the 15 - 20 interval.

Answer:

A. 15 to 20 years old