Answer :
Given the stem-and-leaf plot, we can extract the exact values:
### Step 1: Extracting Data
From the stem-and-leaf plot, we have the following data points:
- 12, 12 (from stem 1, leaves 2, 2)
- 19 (from stem 1, leaf 9)
- 20 (from stem 2, leaf 0)
- 22 (from stem 2, leaf 2)
- 26 (from stem 2, leaf 6)
- 31, 31 (from stem 3, leaves 1, 1)
- 40 (from stem 4, leaf 0)
- 68 (from stem 6, leaf 8)
Thus, the data points are: 12, 12, 19, 20, 22, 26, 31, 31, 40, 68.
### Step 2: Calculate the Mean
To calculate the mean, sum all the values and divide by the number of values:
[tex]\[ \text{Mean} = \frac{\sum \text{data points}}{\text{number of data points}} \][/tex]
[tex]\[ \text{Mean} = \frac{12 + 12 + 19 + 20 + 22 + 26 + 31 + 31 + 40 + 68}{10} \][/tex]
[tex]\[ \text{Mean} = \frac{281}{10} = 28.1 \][/tex]
### Step 3: Calculate the Median
To find the median, we need the middle value(s) in the ordered dataset:
Ordered data: 12, 12, 19, 20, 22, 26, 31, 31, 40, 68.
Since there are 10 data points (an even number), the median will be the average of the 5th and 6th values.
[tex]\[ \text{Median} = \frac{22 + 26}{2} = \frac{48}{2} = 24 \][/tex]
### Step 4: Calculate the Mode
The mode is the value that appears most frequently in the data set. From the data, it appears that 12 and 31 each occur twice.
Thus, there are two modes: 12 and 31.
### Step 5: Calculate the Range
The range is the difference between the maximum and minimum values in the data set.
[tex]\[ \text{Range} = \text{Maximum value} - \text{Minimum value} \][/tex]
[tex]\[ \text{Range} = 68 - 12 = 56 \][/tex]
### Part A: Summary
- Mean: 28.1
- Median: 24
- Mode: 12 and 31 (bimodal)
- Range: 56
### Part B: Analysis
In the context of reporting one measure of central tendency to show they are less prepared for the tournament, the tennis players should consider their objective:
- Mean: Since the mean is skewed higher by the outlier (68 minutes), it does not accurately reflect the central tendency of the majority of players.
- Median: The median is more robust against outliers and provides a better measure of the central location in a skewed distribution.
To present themselves as less prepared, the players should report the median value (24 minutes), as it represents the central tendency more accurately without being influenced by the outlier.
### Conclusion
The tennis players should report the median value to show they are less prepared for the tournament because the median is not affected by the outlier and better represents the overall practice time.
### Step 1: Extracting Data
From the stem-and-leaf plot, we have the following data points:
- 12, 12 (from stem 1, leaves 2, 2)
- 19 (from stem 1, leaf 9)
- 20 (from stem 2, leaf 0)
- 22 (from stem 2, leaf 2)
- 26 (from stem 2, leaf 6)
- 31, 31 (from stem 3, leaves 1, 1)
- 40 (from stem 4, leaf 0)
- 68 (from stem 6, leaf 8)
Thus, the data points are: 12, 12, 19, 20, 22, 26, 31, 31, 40, 68.
### Step 2: Calculate the Mean
To calculate the mean, sum all the values and divide by the number of values:
[tex]\[ \text{Mean} = \frac{\sum \text{data points}}{\text{number of data points}} \][/tex]
[tex]\[ \text{Mean} = \frac{12 + 12 + 19 + 20 + 22 + 26 + 31 + 31 + 40 + 68}{10} \][/tex]
[tex]\[ \text{Mean} = \frac{281}{10} = 28.1 \][/tex]
### Step 3: Calculate the Median
To find the median, we need the middle value(s) in the ordered dataset:
Ordered data: 12, 12, 19, 20, 22, 26, 31, 31, 40, 68.
Since there are 10 data points (an even number), the median will be the average of the 5th and 6th values.
[tex]\[ \text{Median} = \frac{22 + 26}{2} = \frac{48}{2} = 24 \][/tex]
### Step 4: Calculate the Mode
The mode is the value that appears most frequently in the data set. From the data, it appears that 12 and 31 each occur twice.
Thus, there are two modes: 12 and 31.
### Step 5: Calculate the Range
The range is the difference between the maximum and minimum values in the data set.
[tex]\[ \text{Range} = \text{Maximum value} - \text{Minimum value} \][/tex]
[tex]\[ \text{Range} = 68 - 12 = 56 \][/tex]
### Part A: Summary
- Mean: 28.1
- Median: 24
- Mode: 12 and 31 (bimodal)
- Range: 56
### Part B: Analysis
In the context of reporting one measure of central tendency to show they are less prepared for the tournament, the tennis players should consider their objective:
- Mean: Since the mean is skewed higher by the outlier (68 minutes), it does not accurately reflect the central tendency of the majority of players.
- Median: The median is more robust against outliers and provides a better measure of the central location in a skewed distribution.
To present themselves as less prepared, the players should report the median value (24 minutes), as it represents the central tendency more accurately without being influenced by the outlier.
### Conclusion
The tennis players should report the median value to show they are less prepared for the tournament because the median is not affected by the outlier and better represents the overall practice time.
