To determine the shape of the sampling distribution for the difference in the mean salaries ([tex]$\bar{x}_B - \bar{x}_F$[/tex]) of professional baseball and football players, we need to recall the Central Limit Theorem (CLT). The CLT states that when the sample size is sufficiently large (typically [tex]$n > 30$[/tex]), the sampling distribution of the sample mean can be modeled by a Normal distribution regardless of the population distribution’s shape.
Given:
- The sample size for baseball players is 40.
- The sample size for football players is 35.
Both of these sample sizes are greater than 30, hence large enough to apply the Central Limit Theorem.
### Step-by-Step Solution
1. Analyze the Population Distributions:
- The salaries for baseball and football players are heavily skewed right.
- Even though individual populations are skewed, we are looking at the sampling distribution of the sample means.
2. Central Limit Theorem Application:
- Baseball Players: With a sample size of 40, the sampling distribution of the mean salary of baseball players is approximately Normal.
- Football Players: With a sample size of 35, the sampling distribution of the mean salary of football players is approximately Normal.
3. Difference of Means Distribution:
- The difference in sample means [tex]$\bar{x}_B - \bar{x}_F$[/tex] will also follow a Normal distribution since both sampling distributions are approximately Normal.
Therefore, based on the Central Limit Theorem and the given sample sizes, the shape of the sampling distribution for [tex]$\bar{x}_B - \bar{x}_F$[/tex] is approximately Normal. Hence:
### Final Answer:
Approximately Normal since both sample sizes are greater than 30.
