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https://file.aiquickdraw.com/mathbot/file/172703044800868oxiniw.jpg

https://file.aiquickdraw.com/mathbot/file/1727030498955l95g0s22.jpg

Let's break down and solve the questions from the images you have provided step by step.

### a) Better summary of the number of home runs: mean or median?
- The dataset is **skewed**, as seen from the histogram (most players hit fewer home runs, and a few hit many home runs).
- In skewed distributions, **median** is generally a better summary because it is less affected by extreme values.

So, the answer should be:  
- **Median** is a better summary because the **data is skewed**.

### b) Better description of the spread: standard deviation or IQR?
- Since the data is skewed, **IQR (Interquartile Range)** is usually preferred because it measures the spread of the middle 50% of the data and is not as influenced by outliers or skewness, unlike standard deviation.

So, the answer should be:  
- **IQR** is a better summary of the spread because the **data is skewed**.

### c) Using a Normal model, what percentage of players should be within three standard deviations of the mean?
- According to the **Empirical Rule (68-95-99.7 Rule)** for a normal distribution:
  - 68% of data is within 1 standard deviation.
  - 95% is within 2 standard deviations.
  - **99.7% is within 3 standard deviations**.

So, the correct answer is:  
- **99.7%** (Option A).

### d) Using rough estimates, what percentage of players are within three standard deviations of the mean?
- Given the skewed distribution shown in the histogram, it seems that **fewer players** than expected would fall within three standard deviations.
- An estimate of **79%** (Option B) seems reasonable, considering the skewness of the data.

### e) Explain the discrepancy between parts c and d:
- The discrepancy is likely due to the **skewness** of the data. The normal model assumes symmetry, but the distribution of home runs is **skewed to the right** (as seen in the histogram).
- The **data contains outliers** (extremely high home runs for some players).
-

The displayed chart and accompanying histogram show the numbers of home runs hit by all baseball players in a league in the 2021 season. Which would be a better summary of the number of home runs: mean or median? Which is a better description of the spread: standard deviation or IQR? What percentage of players should be within three standard deviations of the mean using a normal model? Why is there a discrepancy between the theoretical and observed percentages?

This problem covers key statistical concepts, including the use of mean and median for skewed data, comparison of standard deviation and IQR as measures of spread, and applying the empirical rule to understand normal distribution. Suitable for high school and college-level students.

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