Sure! Let's analyze the golfer's performance over the two-year period using the given scores for the 2011 and 2012 seasons.
a. Use the mean and standard deviation to evaluate the golfer's performance over the two-year period.
### 2011 Season:
- Scores: 74, 78, 79, 77, 75, 73, 75, 77
- Mean (average) score:
[tex]\[
\text{Mean}_{2011} = 76.0
\][/tex]
- Standard deviation:
[tex]\[
\text{Std}_{2011} = 1.9364916731037085
\][/tex]
### 2012 Season:
- Scores: 71, 70, 75, 77, 85, 80, 71, 79
- Mean (average) score:
[tex]\[
\text{Mean}_{2012} = 76.0
\][/tex]
- Standard deviation:
[tex]\[
\text{Std}_{2012} = 4.924428900898052
\][/tex]
The mean score for both years is 76.0, indicating that the golfer's average performance did not change from 2011 to 2012. However, the standard deviation for 2012 is higher than that for 2011. This indicates that the scores in 2012 were more spread out or variable, showing greater fluctuations in performance compared to 2011.
### b. What is the primary difference in performance between 2011 and 2012? What improvement, if any, can be seen in the 2012 scores?
Primary Difference:
The primary difference between the 2011 and 2012 seasons can be seen in the standard deviation of the scores. The standard deviation for 2012 (4.924428900898052) is significantly higher than that for 2011 (1.9364916731037085). This suggests that while the average score remained the same at 76.0, the golfer's performance was more inconsistent in 2012, with a wider range of scores.
Improvement:
Since the mean score remained unchanged at 76.0, there is no significant improvement in the average score for the two years. However, analyzing the spread of scores might tell another story:
- 2011 Scores Spread: The scores in 2011 are clustered around the mean with a lower standard deviation, indicating more consistent performance.
- 2012 Scores Spread: In 2012, the scores varied more, with both lower lows (e.g., 70, 71) and higher highs (e.g., 85, 80).
The fluctuation towards higher scores (like 85 and 80) in 2012 indicates that the golfer had some impressive rounds, even though the overall inconsistency slightly increased. While the variation suggests inconsistency, these higher scores could point toward potential improvement in skill, albeit sporadically demonstrated.
In conclusion, while there is no improvement in the mean score, the presence of higher individual round scores in 2012 could suggest that the golfer has the potential for better performance, provided they can stabilize their game.
