Sure, let's go through the process step-by-step.
### a. Use the mean and standard deviation to evaluate the golfer's performance over the two-year period.
#### 2011 Season
1. Calculate the Mean Score:
The mean score for the 2011 season is calculated by summing all the scores and dividing by the number of scores. The mean score for 2011 is:
[tex]\[
\text{Mean}_{2011} = 76.0
\][/tex]
2. Calculate the Standard Deviation:
The standard deviation provides a measure of the variation or spread of the scores from the mean. For the 2011 season, the standard deviation is:
[tex]\[
\text{STD}_{2011} = 2.070
\][/tex]
#### 2012 Season
1. Calculate the Mean Score:
The mean score for the 2012 season is:
[tex]\[
\text{Mean}_{2012} = 76.0
\][/tex]
2. Calculate the Standard Deviation:
The standard deviation for the 2012 season is:
[tex]\[
\text{STD}_{2012} = 5.264
\][/tex]
### b. What is the primary difference in performance between 2011 and 2012? What improvement, if any, can be seen in the 2012 scores?
1. Difference in Mean Scores:
The difference in mean scores between the two years is calculated as:
[tex]\[
\text{Mean Difference} = \text{Mean}_{2012} - \text{Mean}_{2011} = 76.0 - 76.0 = 0.0
\][/tex]
This indicates that the average performance in terms of scoring remained consistent between 2011 and 2012.
2. Difference in Standard Deviations:
The difference in standard deviations between the two years is:
[tex]\[
\text{STD Difference} = \text{STD}_{2012} - \text{STD}_{2011} = 5.264 - 2.070 = 3.194
\][/tex]
This indicates that there was a higher variability in scores during the 2012 season compared to the 2011 season. In 2011, the scores were more consistent, whereas, in 2012, there was a wider range of scores.
3. Improvement Evaluation:
In golf, a lower mean score indicates better performance. Since the mean score for 2012 is equal to that of 2011, there is no improvement in terms of average score. An improvement would have been indicated if the mean score for 2012 was lower than that of 2011. The improvement score can be formally calculated as:
[tex]\[
\text{Improvement} = \max(0, \text{Mean}_{2011} - \text{Mean}_{2012}) = \max(0, 76.0 - 76.0) = 0
\][/tex]
### Summary
- Both years have the same average score (76.0).
- The scores in 2011 were more consistent (lower standard deviation of 2.070) compared to 2012 (higher standard deviation of 5.264).
- There was no improvement in the golfer's average score from 2011 to 2012.
