Question

select the correct answer.
the data set {7, 20, 51, 6, 30, 72, 31, 84, 28, 77, 98} is to be represented by a histogram. which first interval would most clearly and simply show the distribution of this data?

a. 0 - 5
b. 0 - 10
c. 0 - 50
d. 0 - 100

Explanation:

Step1: Analyze the data set

The data set is {7, 20, 51, 6, 30, 72, 31, 84, 28, 77, 98}. The minimum value is 6 and the maximum is 98.

Step2: Evaluate each option

- Option A (0 - 5): The data has no values in this interval (minimum is 6), so it's not useful.
- Option B (0 - 10): Values 6 and 7 are in this interval, but other values (like 20, 30, etc.) are outside. It may not show the distribution clearly as many values are in higher ranges.
- Option C (0 - 50): Values 7, 20, 51? Wait, 51 is just above 50. Wait, 7, 20, 6, 30, 28, 31 are in 0 - 50 (51 is 51, so close). But let's check Option D.
- Option D (0 - 100): All values (7, 20, 51, 6, 30, 72, 31, 84, 28, 77, 98) are within 0 - 100. But wait, the question is about the first interval. Wait, maybe I misread. Wait, the first interval for a histogram. Wait, maybe the intervals are consecutive. Wait, no, the question is about which first interval (the starting interval) would most clearly show the distribution. Wait, maybe the intervals are like 0 - 10, 10 - 20, etc. But the options are single intervals. Wait, no, the options are possible first intervals. Wait, let's re - evaluate.

Wait, the data has values from 6 to 98. Option A: 0 - 5 has no data. Option B: 0 - 10 has 6 and 7. Option C: 0 - 50 has 7, 20, 6, 30, 28, 31, 51? No, 51 is outside 0 - 50. Wait, 51 is 51, so 0 - 50 would include up to 50. So 51 is outside. Option D: 0 - 100 includes all data. But a histogram with a single interval 0 - 100 would just be a single bar, which doesn't show distribution. Wait, maybe the question is about the class interval width. Wait, maybe I made a mistake. Wait, let's check the data again. The values are 7, 20, 51, 6, 30, 72, 31, 84, 28, 77, 98.

Wait, maybe the first interval is the left - most interval. Let's see the minimum value is 6. Option A: 0 - 5: no data. Option B: 0 - 10: data points 6 and 7. Option C: 0 - 50: data points 6, 7, 20, 28, 30, 31, 51? No, 51 is 51, so 0 - 50 is up to 50, so 51 is outside. Option D: 0 - 100: all data. But a histogram with a single interval 0 - 100 is not useful. Wait, maybe the question is about the interval that can be the first in a set of intervals. Wait, maybe the intervals are like 0 - 10, 10 - 20, 20 - 30, etc. But the options are given as single intervals. Wait, perhaps the question is misphrased, and it's about the class interval that is appropriate for the first bin.

Wait, let's think again. The data has values from 6 to 98. Option A: 0 - 5: no data. Option B: 0 - 10: has two values. Option C: 0 - 50: has most of the lower - valued data (6, 7, 20, 28, 30, 31) and 51 is just above 50. Option D: 0 - 100: all data, but it's too broad. But wait, maybe the question is about the interval that includes the minimum value and starts at 0. The minimum value is 6, so 0 - 10 includes 6 and 7. But 0 - 50 includes more data (6,7,20,28,30,31,51? No, 51 is 51, so 0 - 50 is up to 50, so 51 is outside). Wait, maybe the question is about the interval that is the most appropriate to start with to show the distribution. Wait, maybe I made a mistake in Option C. Wait, 51 is 51, so 0 - 50: 51 is not in it. But 7, 20, 6, 30, 28, 31 are in 0 - 50. Option D: 0 - 100 includes all, but it's too wide. Wait, the question says "most clearly and simply show the distribution". If we take 0 - 10 as the first interval, then the next intervals would be 10 - 20, 20 - 30, etc. But the options are single intervals. Wait, maybe the question is about which interval (as the first one) contains the most relevant data. Wait, no. Wait, let's check the values again. The data points: 6,7 (in 0 - 10), 20,28 (in 10 - 30), 30,31 (in 30 - 40), 51 (50 - 60), 72,77 (70 - 80), 84 (80 - 90), 98 (90 - 100). If the first interval is 0 - 10, it has two points. If it's 0 - 50, it has 6,7,20,28,30,31 (and 51 is just outside). But 0 - 100 has all, but it's not a good interval for showing distribution. Wait, maybe the question is a trick question. Wait, the minimum value is 6, so 0 - 5 has no data. 0 - 10 has 6 and 7. 0 - 50 has more data (excluding 51,72,77,84,98). But 0 - 100 has all. But the question is about the first interval. Wait, maybe the answer is B? No, wait, no. Wait, maybe I misread the options. Wait, the options are A. 0 - 5, B. 0 - 10, C. 0 - 50, D. 0 - 100.

Wait, the key is that the first interval should include the minimum value and be such that it starts at 0 (as a common starting point for histograms) and shows the data. Since the minimum is 6, 0 - 10 includes 6 and 7. But 0 - 50 includes more of the lower - valued data. But 51 is 51, which is just above 50. Wait, maybe the question is about the interval that is the most appropriate to start with, considering that the data has values from 6 to 98. Wait, 0 - 100 is too broad, 0 - 5 has no data, 0 - 10 has two data points, 0 - 50 has 6,7,20,28,30,31 (and 51 is 51, so close). But maybe the correct answer is B? No, wait, no. Wait, let's check the number of data points in each interval:

- A: 0 - 5: 0 points.
- B: 0 - 10: 2 points (6,7).
- C: 0 - 50: 6,7,20,28,30,31 (6 points, 51 is outside).
- D: 0 - 100: 11 points (all).

But a histogram with a single interval (0 - 100) doesn't show distribution. But the question is about the first interval. Maybe the intervals are supposed to be like 0 - 10, 10 - 20, etc. So the first interval is 0 - 10. But in the options, the first interval is one of these. Wait, maybe the question is asking which interval (as the starting interval) would allow for a clear distribution. If we take 0 - 10 as the first interval, then the next intervals can be 10 - 20, 20 - 30, etc., which can show the distribution. If we take 0 - 50 as the first interval, the next interval would be 50 - 100, which would group too many values. So the best first interval is 0 - 10? No, wait, 0 - 10 has only two values. Wait, maybe I made a mistake. Wait, the data set: 7, 20, 51, 6, 30, 72, 31, 84, 28, 77, 98. Let's count the number of values in each option's interval:

- A: 0 - 5: 0
- B: 0 - 10: 6,7 → 2 values
- C: 0 - 50: 7,20,6,30,28,31 → 6 values (51 is 51, so not in 0 - 50)
- D: 0 - 100: 11 values

But the question is about "most clearly and simply show the distribution". A histogram with a first interval of 0 - 10 will have a bar with 2 values, then 10 - 20 (20 is in it, so 1 value), 20 - 30 (20,28 → 2 values), 30 - 40 (30,31 → 2 values), 40 - 50 (0), 50 - 60 (51 → 1), 60 - 70 (0), 70 - 80 (72,77 → 2), 80 - 90 (84 → 1), 90 - 100 (98 → 1). This shows the distribution. If the first interval is 0 - 50, the first bar would have 6 values (excluding 51), then 50 - 100 would have 5 values (51,72,77,84,98). This groups too much in the second interval. If the first interval is 0 - 100, it's just one bar. So the best first interval is 0 - 10? But wait, the minimum value is 6, so 0 - 10 includes the lower values. But earlier I thought 0 - 50 has more. Wait, no, the key is that the first interval should be such that it starts at 0 (a natural starting point) and includes the minimum value, and allows for subsequent intervals to show the distribution. Since the minimum is 6, 0 - 10 includes it, and subsequent intervals (10 - 20, 20 - 30, etc.) can be created to show the distribution. 0 - 5 has no data, 0 - 50 groups too many in the first interval, and 0 - 100 is too broad. So the correct answer is B? Wait, no, wait the option B is 0 - 10. But let's check again. Wait, the data points: 6 and 7 are in 0 - 10, 20 is in 10 - 20, 28 is in 20 - 30, 30 and 31 are in 30 - 40, 51 is in 50 - 60, 72 and 77 are in 70 - 80, 84 is in 80 - 90, 98 is in 90 - 100. So with first interval 0 - 10, we can have a histogram with intervals 0 - 10, 10 - 20, 20 - 30, 30 - 40, 40 - 50, 50 - 60, 60 - 70, 70 - 80, 80 - 90, 90 - 100, which shows the distribution clearly. If we take first interval 0 - 50, the next interval is 50 - 100, which has 5 data points, which is not as clear. So the correct answer is B. 0 - 10. Wait, but earlier I thought 0 - 50 has more. No, the key is the ability to show the distribution, not the number of data points in the first interval. The first interval should be narrow enough to start showing the distribution, and 0 - 10 is narrow enough to include the lower values and allow subsequent intervals to be created.

Answer:

B. 0 - 10