To determine whether the beginner is performing at the average level, we need to calculate the average distance of the hits. The average distance can be computed using the midpoint of each distance interval and weighting these midpoints by their respective frequencies.
Here's a detailed step-by-step solution:
1. List out the distance intervals and frequencies:
- [tex]\( 135 \leq d < 140 \)[/tex]: 4 hits
- [tex]\( 140 \leq d < 145 \)[/tex]: 13 hits
- [tex]\( 145 \leq d < 150 \)[/tex]: 17 hits
- [tex]\( 150 \leq d < 155 \)[/tex]: 6 hits
2. Calculate the midpoint of each distance interval:
- For [tex]\( 135 \leq d < 140 \)[/tex]: The midpoint is [tex]\( \frac{135 + 140}{2} = 137.5 \)[/tex] yards.
- For [tex]\( 140 \leq d < 145 \)[/tex]: The midpoint is [tex]\( \frac{140 + 145}{2} = 142.5 \)[/tex] yards.
- For [tex]\( 145 \leq d < 150 \)[/tex]: The midpoint is [tex]\( \frac{145 + 150}{2} = 147.5 \)[/tex] yards.
- For [tex]\( 150 \leq d < 155 \)[/tex]: The midpoint is [tex]\( \frac{150 + 155}{2} = 152.5 \)[/tex] yards.
3. Calculate the total number of hits (the sum of frequencies):
[tex]\[
4 + 13 + 17 + 6 = 40
\][/tex]
So, the total frequency is 40.
4. Calculate the weighted average distance:
We multiply each midpoint by its corresponding frequency and sum those products, then divide by the total number of hits.
[tex]\[
\text{Average distance} = \frac{(4 \times 137.5) + (13 \times 142.5) + (17 \times 147.5) + (6 \times 152.5)}{40}
\][/tex]
Let's calculate the products:
[tex]\[
4 \times 137.5 = 550
\][/tex]
[tex]\[
13 \times 142.5 = 1852.5
\][/tex]
[tex]\[
17 \times 147.5 = 2507.5
\][/tex]
[tex]\[
6 \times 152.5 = 915
\][/tex]
Now, sum these products:
[tex]\[
550 + 1852.5 + 2507.5 + 915 = 5825
\][/tex]
Finally, divide by the total number of hits:
[tex]\[
\text{Average distance} = \frac{5825}{40} = 145.625 \text{ yards}
\][/tex]
5. Compare the calculated average distance with the coach's specified goal:
The coach mentioned that the beginner should be able to hit the ball 150 yards on average. We calculated that the beginner's average distance is [tex]\( 145.625 \)[/tex] yards.
Conclusion:
The beginner is not hitting the ball exactly 150 yards on average. Instead, the average distance of the beginner's hits is [tex]\( 145.625 \)[/tex] yards, which is below the coach's target of 150 yards. Therefore, the beginner is not performing at the expected average level.
