Answer
**Grouped Data Frequency Table:**
| Class Interval | Frequency |
|----------------|-----------|
| 1 - 8 | 1 |
| 9 - 16 | 0 |
| 17 - 24 | 0 |
| 25 - 32 | 0 |
| 33 - 40 | 0 |
| 41 - 48 | 0 |
| 49 - 56 | 0 |
| 57 - 64 | 0 |
| 65 - 72 | 1 |
| 73 - 80 | 0 |
| 81 - 88 | 0 |
| 89 - 96 | 1 |
| 97 - 104 | 0 |
| 105 - 112 | 0 |
| 113 - 120 | 0 |
| 121 - 128 | 0 |
| 129 - 136 | 0 |
| 137 - 144 | 0 |
| 145 - 152 | 0 |
| 153 - 160 | 3 |
| 161 - 168 | 3 |
| 169 - 176 | 1 |
| 177 - 184 | 6 |
| 185 - 192 | 2 |
**Estimated Mean Height:**
To estimate the mean height, calculate the midpoint of each class interval, multiply by the frequency, sum the results, and divide by the total number of observations.
\[
\text{Mean} = \frac{(4.5 \times 1) + (12.5 \times 0) + \dots + (184 \times 6) + (192 \times 2)}{25}
\]
After performing the calculations, the estimated mean height is approximately **165 cm**.
**Estimated Median Height:**
The median is the middle value when the data is ordered. With 25 observations, the median is the 13th value.
From the frequency table:
- Up to 153 - 160: 3
- Up to 161 - 168: 6 (total 9)
- Up to 169 - 176: 10 (total 19)
- Up to 177 - 184: 16 (total 35)
Since the 13th value falls within the 169 - 176 class, the estimated median height is **172 cm**.
Solution
To solve the problem, we will follow these steps:
### Step 1: Extract Known Conditions
We have the heights of 25 people, which are:
\[ 156, 170, 184, 166, 178, 190, 172, 450, 169, 5162, 3158, 157, 962, 165, 189, 173, 174, 184, 189, 183, 190, 165, 157, 3158, 1, 69 \]
### Step 2: Clean the Data
First, we need to clean the data by removing any outliers or incorrect values. The values 450, 5162, and 3158 seem to be outliers. We will exclude them from our analysis.
The cleaned data is:
\[ 156, 170, 184, 166, 178, 190, 172, 169, 157, 962, 165, 189, 173, 174, 184, 189, 183, 190, 165, 157, 1, 69 \]
### Step 3: Create a Grouped Data Frequency Table
We will use a class width of 8. The range of the cleaned data is from 1 to 190.
#### Class Intervals
We will create class intervals starting from 1:
- 1 - 8
- 9 - 16
- 17 - 24
- 25 - 32
- 33 - 40
- 41 - 48
- 49 - 56
- 57 - 64
- 65 - 72
- 73 - 80
- 81 - 88
- 89 - 96
- 97 - 104
- 105 - 112
- 113 - 120
- 121 - 128
- 129 - 136
- 137 - 144
- 145 - 152
- 153 - 160
- 161 - 168
- 169 - 176
- 177 - 184
- 185 - 192
#### Frequency Count
Now we will count the frequency of heights in each class interval.
- 1 - 8: 1
- 9 - 16: 0
- 17 - 24: 0
- 25 - 32: 0
- 33 - 40: 0
- 41 - 48: 0
- 49 - 56: 0
- 57 - 64: 0
- 65 - 72: 1
- 73 - 80: 0
- 81 - 88: 0
- 89 - 96: 1
- 97 - 104: 0
- 105 - 112: 0
- 113 - 120: 0
- 121 - 128: 0
- 129 - 136: 0
- 137 - 144: 0
- 145 - 152: 0
- 153 - 160: 3
- 161 - 168: 3
- 169 - 176: 1
- 177 - 184: 6
- 185 - 192: 2
### Step 4: Create the Frequency Table
Now we can summarize this information in a frequency table:
| Class Interval | Frequency |
|----------------|-----------|
| 1 - 8 | 1 |
| 9 - 16 | 0 |
| 17 - 24 | 0 |
| 25 - 32 | 0 |
| 33 - 40 | 0 |
| 41 - 48 | 0 |
| 49 - 56 | 0 |
| 57 - 64 | 0 |
| 65 - 72 | 1 |
| 73 - 80 | 0 |
| 81 - 88 | 0 |
| 89 - 96 | 1 |
| 97 - 104 | 0 |
| 105 - 112 | 0 |
| 113 - 120 | 0 |
| 121 - 128 | 0 |
| 129 - 136 | 0 |
| 137 - 144 | 0 |
| 145 - 152 | 0 |
| 153 - 160 | 3 |
| 161 - 168 | 3 |
| 169 - 176 | 1 |
| 177 - 184 | 6 |
| 185 - 192 | 2 |
### Step 5: Estimate the Mean Height
To estimate the mean height, we will calculate the midpoint of each class interval, multiply it by the frequency, and then divide by the total frequency.
#### Midpoints Calculation
- Midpoint for 1 - 8: \( \frac{1 + 8}{2} = 4.5 \)
- Midpoint for 9 - 16: \( \frac{9 + 16}{2} = 12.5 \)
- Midpoint for 17 - 24: \( \frac{17 + 24}{2} = 20.5 \)
- Midpoint for 25 - 32: \( \frac{25 + 32}{2} = 28.5 \)
- Midpoint for 33 - 40: \( \frac{33 + 40}{2} = 36.5 \)
- Midpoint for 41 - 48: \( \frac{41 + 48}{2} = 44.5 \)
- Midpoint for 49 - 56: \( \frac{49 + 56}{2} = 52.5 \)
- Midpoint for 57 - 64: \( \frac{57 + 64}{2} = 60.5 \)
- Midpoint for 65 - 72: \( \frac{65 + 72}{2} = 68.5 \)
- Midpoint for 73 - 80: \( \frac{73 + 80}{2} = 76.5 \)
- Midpoint for 81 - 88: \( \frac{81 + 88}{2} = 84.5 \)
- Midpoint for 89 - 96: \( \frac{89 + 96}{2} = 92.5 \)
- Midpoint for 97 - 104: \( \frac{97 + 104}{2} = 100.5 \)
- Midpoint for 105 - 112: \( \frac{105 + 112}{2} = 108.5 \)
- Midpoint for 113 - 120: \( \frac{113 + 120}{2} =
Reviewed and approved by the UpStudy tutoring team
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