Certainly, let's go through the problem step by step.
### Scores from the First 10 Matches:
Salary scored the following points in her first 10 matches:
[tex]\[ 45, 36, 50, 27, 36, 52, 50, 43, 50, 47 \][/tex]
### Step 1: Most Frequent Score Point
To find the most frequent score, we count how many times each score appears:
- 50 appears 3 times
- 36 appears 2 times
- 45, 27, 52, 43, and 47 each appear 1 time
The score that appears the most frequently is 50.
### Step 2: Mean and Median of the Initial 10 Matches
#### Mean
To calculate the mean (average) score:
[tex]\[ \text{Mean} = \frac{45 + 36 + 50 + 27 + 36 + 52 + 50 + 43 + 50 + 47}{10} \][/tex]
[tex]\[ \text{Mean} = \frac{436}{10} \][/tex]
[tex]\[ \text{Mean} = 43.6 \][/tex]
#### Median
To find the median, we need to order the scores and find the middle value (or the average of the two middle values if there's an even number of scores):
Ordered scores: [tex]\[ 27, 36, 36, 43, 45, 47, 50, 50, 50, 52 \][/tex]
Since there are 10 scores, the median is the average of the 5th and 6th scores:
[tex]\[ \text{Median} = \frac{45 + 47}{2} = \frac{92}{2} = 46 \][/tex]
So, the mean score of the first 10 matches is 43.6 and the median score is 46.
### Step 3: Comparing Mean and Median
From our calculations:
- Mean [tex]\( \approx 43.6 \)[/tex]
- Median [tex]\( = 46 \)[/tex]
Since 43.6 is not equal to 46, the mean and median are not equal.
### Step 4: Change in Mean Score After the 11th Match
Salary scored 56 points in her 11th match. Let's first append this new score to the list:
[tex]\[ 45, 36, 50, 27, 36, 52, 50, 43, 50, 47, 56 \][/tex]
#### Mean After the 11th Match
To calculate the new mean:
[tex]\[ \text{New Mean} = \frac{45 + 36 + 50 + 27 + 36 + 52 + 50 + 43 + 50 + 47 + 56}{11} \][/tex]
[tex]\[ \text{New Mean} \approx \frac{491}{11} \][/tex]
[tex]\[ \text{New Mean} \approx 44.727 \][/tex]
#### Change in Mean
The change in the mean is the difference between the new mean and the original mean:
[tex]\[ \text{Change in Mean} \approx 44.727 - 43.6 \][/tex]
[tex]\[ \text{Change in Mean} \approx 1.127 \][/tex]
### Summary
1. Most Frequent Score Point: 50
2. Mean and Median Equality: No, the mean (43.6) and median (46) are not equal.
3. Change in Mean Score After the 11th Match: The mean score increased by approximately 1.127 points.
