Let's analyze the given data step-by-step:
Scores: 65, 72, 70, 69, 72, 67, 70, 72, 73
### a. Find the median score.
The median is the middle value in an ordered list. Let's first order the scores:
67, 65, 69, 70, 70, 72, 72, 72, 73
Now:
65, 67, 69, 70, 70, 72, 72, 72, 73
Since the number of scores is odd (9), the median is the middle value:
[tex]\[ \text{Median} = 70 \][/tex]
### b. Find the mode of the scores.
The mode is the most frequently occurring value. Let's count the frequency of each score:
- 65: 1
- 67: 1
- 69: 1
- 70: 2
- 72: 3
- 73: 1
The mode is the score that appears the most often:
[tex]\[ \text{Mode} = 72 \][/tex]
### c. Find the range of the set of data.
The range is the difference between the highest and lowest scores:
[tex]\[ \text{Range} = \text{Max score} - \text{Min score} = 73 - 65 = 8 \][/tex]
### d. Find the mean of the set of data.
The mean is the sum of all scores divided by the number of scores:
[tex]\[ \text{Mean} = \frac{(65 + 72 + 70 + 69 + 72 + 67 + 70 + 72 + 73)}{9} \][/tex]
First, sum the scores:
[tex]\[ 65 + 72 + 70 + 69 + 72 + 67 + 70 + 72 + 73 = 630 \][/tex]
Now divide by the number of scores (9):
[tex]\[ \text{Mean} = \frac{630}{9} = 70 \][/tex]
### e. Another player scores 80. Predict how this player's score will change the median, mode, range, and mean of the data and explain your reasoning. Then compute each of these measures to check your predictions.
Let's add the new score (80) to the list and analyze the new data:
65, 72, 70, 69, 72, 67, 70, 72, 73, 80
Order the new list:
65, 67, 69, 70, 70, 72, 72, 72, 73, 80
We now have 10 scores.
#### New Median:
With an even number of scores (10), the median is the average of the 5th and 6th scores:
[tex]\[ \text{New Median} = \frac{70 + 72}{2} = \frac{142}{2} = 71 \][/tex]
#### New Mode:
The mode is still the most frequent value in the dataset:
- 65: 1
- 67: 1
- 69: 1
- 70: 2
- 72: 3
- 73: 1
- 80: 1
The mode remains:
[tex]\[ \text{Mode} = 72 \][/tex]
#### New Range:
The range will change because the new maximum score is now 80:
[tex]\[ \text{New Range} = 80 - 65 = 15 \][/tex]
#### New Mean:
Sum the new scores:
[tex]\[ 65 + 67 + 69 + 70 + 70 + 72 + 72 + 72 + 73 + 80 = 710 \][/tex]
Now divide by the number of scores (10):
[tex]\[ \text{New Mean} = \frac{710}{10} = 71 \][/tex]
### Summary:
- New Median: 71
- New Mode: 72
- New Range: 15
- New Mean: 71
