One​ season, baseball player A hit 6464 home runs. During a different​ season, baseball player B hit 5757 home runs. How does the performance of player A compare to the performance of player​ B? The table available below shows the means and standard deviations for the number of home runs scored by all players with at least 502 plate appearances in their respective seasons. Use these to determine whose home run feat was more impressive.Season with
Player A
Season with
Player B
Mean
18.318.3
20.720.7
Standard deviation
12.7712.77
13.4313.43

The​ z-score for player​ A, z Subscript Aequals= enter your response here​, is
▼
lessless
greatergreater
than the​ z-score for player​ B, z Subscript Bequals= enter your response here​, so the feat by player
▼
Upper BB
Upper AA
was more impressive

To compare the performances of Player A and Player B, we can use **z-scores**, which help standardize their home run counts based on the mean and standard deviation of their respective seasons.

The **z-score** is calculated using the formula:

\[
z = \frac{{X - \mu}}{{\sigma}}
\]

Where:
- \(X\) is the observed value (the number of home runs hit),
- \(\mu\) is the mean of the population (mean home runs for that season),
- \(\sigma\) is the standard deviation of the population (standard deviation of home runs for that season).

### For Player A:
- \(X_A = 64\) (Player A's home runs),
- \(\mu_A = 18.3\) (mean home runs in Player A's season),
- \(\sigma_A = 12.77\) (standard deviation in Player A's season).

So, the z-score for Player A is:

\[
z_A = \frac{{64 - 18.3}}{{12.77}} = \frac{{45.7}}{{12.77}} \approx 3.58
\]

### For Player B:
- \(X_B = 57\) (Player B's home runs),
- \(\mu_B = 20.7\) (mean home runs in Player B's season),
- \(\sigma_B = 13.43\) (standard deviation in Player B's season).

So, the z-score for Player B is:

\[
z_B = \frac{{57 - 20.7}}{{13.43}} = \frac{{36.3}}{{13.43}} \approx 2.70
\]

### Conclusion:
- The **z-score for Player A** (\(z_A \approx 3.58\)) is **greater** than the **z-score for Player B** (\(z_B \approx 2.70\)).
- Therefore, Player A's home run feat was **more impressive** than Player B's based on their z-scores.

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Do you want more details or have any questions about this?

Here are 5 related questions:
1. What does a higher z-score indicate in terms of performance?
2. How would the comparison change if the standard deviations were larger?
3. What other statistical measures can be used to compare performances?
4. How does using z-scores help in comparing players from different seasons?
5. Can the z-score approach be used to compare other statistics like RBIs or batting averages?

**Tip:** Z-scores are especially useful when comparing performances across different seasons, as they account for varying averages and dispersions in the data.

One season, baseball player A hit 64 home runs. During a different season, baseball player B hit 57 home runs. How does the performance of player A compare to the performance of player B? The table available below shows the means and standard deviations for the number of home runs scored by all players with at least 502 plate appearances in their respective seasons. Use these to determine whose home run feat was more impressive. Season with Player A: Mean 18.3, Standard deviation 12.77. Season with Player B: Mean 20.7, Standard deviation 13.43.

Learn how to compare the home run performances of two baseball players using z-scores, a statistical method that takes into account the mean and standard deviation of their respective seasons. The problem includes calculating z-scores and interpreting results to determine which player's performance was more impressive.

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