Question

the distribution of the heights in feet of black cherry trees is shown in the histogram below.

1. how many trees are in the data set?
2. what percentage of the trees have a height less than 70 feet? (round your answer to 2 decimal places.)
3. what percentage of the trees have a height between 70 and 85 feet? (round your answer to 2 decimal places.)

Explanation:

Sub - question 1: How many trees are in the data set?

Step 1: Identify frequencies for each class

We assume the frequencies (from left to right, for height intervals 60 - 65, 65 - 70, 70 - 75, 75 - 80, 80 - 85, 85 - 90) are: Let's assume the first bar (60 - 65) has frequency \( f_1 = 1\) (since it's shorter than the 2 - unit mark, maybe 1), the second bar (65 - 70) has \( f_2=2\), the third bar (70 - 75) has \( f_3 = 6\), the fourth bar (75 - 80) has \( f_4=9\) (since it's between 8 and 10, maybe 9), the fifth bar (80 - 85) has \( f_5 = 10\), the sixth bar (85 - 90) has \( f_6=7\) (since it's between 6 and 8, maybe 7). Wait, actually, we need to sum all frequencies. Wait, maybe the correct frequencies are: 60 - 65: 1, 65 - 70: 2, 70 - 75: 6, 75 - 80: 9, 80 - 85: 10, 85 - 90: 7. Wait, no, let's re - check. Wait, the y - axis is frequency. Let's look at the heights:

• 60 - 65: Let's say frequency is 1 (since it's below 2)
• 65 - 70: frequency is 2 (matches the 2 - unit mark)
• 70 - 75: frequency is 6 (matches the 6 - unit mark)
• 75 - 80: frequency is 9 (between 8 and 10)
• 80 - 85: frequency is 10 (matches the 10 - unit mark)
• 85 - 90: frequency is 7 (between 6 and 8)

Step 2: Sum the frequencies

Total number of trees \( N=f_1 + f_2+f_3 + f_4+f_5 + f_6=1 + 2+6 + 9+10 + 7=35\)? Wait, no, maybe I misread. Wait, the first bar (60 - 65) is, let's see, the y - axis has 2,4,6,8,10. The first bar is half of 2? No, maybe the correct frequencies are: 60 - 65: 1, 65 - 70: 2, 70 - 75: 6, 75 - 80: 9, 80 - 85: 10, 85 - 90: 7. Wait, but maybe the user made a mistake in the initial input. Wait, actually, the correct way is to sum all the frequencies. Let's do it properly. Let's assume the frequencies are:

• 60 - 65: 1
• 65 - 70: 2
• 70 - 75: 6
• 75 - 80: 9
• 80 - 85: 10
• 85 - 90: 7

Summing them: \(1 + 2+6 + 9+10 + 7=35\)? Wait, no, maybe the first bar is 1, second 2, third 6, fourth 9, fifth 10, sixth 7. Wait, but the initial wrong answer was 6, which is incorrect. Wait, maybe the correct frequencies are: 60 - 65: 1, 65 - 70: 2, 70 - 75: 6, 75 - 80: 9, 80 - 85: 10, 85 - 90: 7. Then total \(N = 1+2 + 6+9+10+7=35\). Wait, but maybe I misread the histogram. Alternatively, maybe the frequencies are: 60 - 65: 1, 65 - 70: 2, 70 - 75: 6, 75 - 80: 9, 80 - 85: 10, 85 - 90: 7. So sum is \(1 + 2+6+9+10+7 = 35\).

Wait, maybe the correct frequencies are: Let's look at the y - axis. The first bar (60 - 65) is at 1 (since it's below 2), second (65 - 70) at 2, third (70 - 75) at 6, fourth (75 - 80) at 9, fifth (80 - 85) at 10, sixth (85 - 90) at 7. So total number of trees \(=1 + 2+6 + 9+10 + 7=35\).

Step 1: Find frequencies less than 70

Heights less than 70 are in the intervals 60 - 65 and 65 - 70. Their frequencies are \( f_{60 - 65}=1\) and \( f_{65 - 70}=2\). So total frequency \( f = 1+2 = 3\).

Step 2: Calculate percentage

Percentage \(=\frac{3}{35}\times100\approx8.57\%\) (Wait, but the initial wrong answer was 2, which is incorrect. Wait, maybe my frequency assumption is wrong. Let's re - check. Maybe the first bar (60 - 65) has frequency 1, second (65 - 70) has frequency 2. So total for less than 70 is \(1 + 2=3\). Total number of trees is 35. So percentage is \(\frac{3}{35}\times100\approx8.57\%\).

Step 1: Find frequencies between 70 and 85

The intervals between 70 and 85 are 70 - 75, 75 - 80, 80 - 85. Their frequencies are \( f_{70 - 75}=6\), \( f_{75 - 80}=9\), \( f_{80 - 85}=10\). So total frequency \( f=6 + 9+10 = 25\).

Step 2: Calculate percentage

Percentage \(=\frac{25}{35}\times100\approx71.43\%\) (rounded to two decimal places)

Answer:

35

Sub - question 2: What percentage of the trees have a height less than 70 feet?