1 Test If There Is A Difference Between The True Average Number Of P ✓ Solved

1) Test if there is a difference between the true average number of plate appearances by different salary tiers (where players making at least 10 million dollars per year are in one tier, those making at least 1 million dollars per year but less than 10 million dollars per year another, and those making less than 1 million dollars per year are in the bottom tier). Hypotheses Assumption Check The samples are categorized in only one way and are independent. We should check to see if they have similar variances and distributions that are approximately normal to see if our assumptions hold. (Those these last requirements only need to be loosely met.) Comment by Dr. Ryan Sides: This is not done for you here, but as this was part of previous assignments, make sure you check this and show whether or not your assumptions truly hold.

Decision Rule If the p-value is less than the pre-determined alpha of 0.05, we will reject the null hypothesis. Descriptive Statistics p-value Decision Because the p-value of 0.0132 is less than a pre-determined alpha of 0.05, we reject the null hypothesis. Conclusion We can conclude that at least one population mean number of plate appearances is different by salary tier when considering MLB players in 2008 who had at least 200 plate appearances. Pairwise Comparisons Of the three pairwise comparisons that can be made among the groups, we show statistical significance between both the filthy rice and not as rich players as well as the filthy rich and very rich players. Thus, it can be concluded that the filthy rich players actually have an average number of plate appearances different from the other two groups.

2) Test if there is a difference between the actual mean number of home runs hit by players in different speed categories (where players with at least 20 stolen bases are in one category, those with at least 10 stolen bases but less than 20 another, those with at least 5 stolen bases but less than 10 another, and those with less than 5 stolen bases are in the bottom category). Hypotheses Assumption Check The samples are categorized in only one way and are independent. We should check to see if they have similar variances and distributions that are approximately normal to see if our assumptions hold, those these last requirements only need to be loosely met. Decision Rule If the p-value is less than the pre-determined alpha of 0.05, we will reject the null hypothesis.

Descriptive Statistics p-value Decision Because the p-value of 0.374 is greater than a pre-determined alpha of 0.05, we fail to reject the null hypothesis. Conclusion We cannot conclude that at least one population mean number of home runs hit is different by speed category when considering MLB players in 2008 who had at least 200 plate appearances. Code: Criteria Exemplary 20 points Accomplished 17 points Developing 15 points Beginning 13 points Criterion Score Focus/Thesis Student exhibits a defined and clear understanding of the assignment. Thesis is clearly defined and well-constructed to help guide the reader throughout the assignment. Student builds upon the thesis of the assignment with well-documented and exceptional supporting facts, figures, and/or statements.

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Paper for above instructions

Statistical Analysis of MLB Player Performance Based on Salary and Speed Categories

Statistical analysis is a fundamental aspect of sports analytics, allowing researchers and professionals to draw meaningful conclusions from raw data. In this analysis, we will test whether there is a difference in the mean plate appearances of Major League Baseball (MLB) players based on salary tiers, and whether the actual mean number of home runs hit varies with stolen base categories. The analysis will use inferential statistics to arrive at conclusions and will include the necessary checks for assumptions regarding the data.

Test 1: Plate Appearances Based on Salary Tiers

Hypotheses

- Null Hypothesis (H0): The mean number of plate appearances is the same across all salary tiers.
- Alternative Hypothesis (H1): At least one salary tier has a different mean number of plate appearances.

Assumption Check

Before conducting statistical tests, we must check key assumptions that are critical for a valid analysis. The samples of the plate appearances are categorized into three tiers:
1. Tier 1: Players making at least million/year
2. Tier 2: Players making between million and million/year
3. Tier 3: Players making less than million/year
The main assumptions are:
1. Independence: Each tier is independent of one another.
2. Normality: The distributions of plate appearances in each group should be roughly normal.
3. Homogeneity of Variance: The variances among the different salary tiers should be approximately equal.
Using the Levene’s test for homogeneity of variances, we would want to confirm that the p-value is above 0.05, indicating that variances among the groups are equal. Additionally, Shapiro-Wilk tests can verify normality in each group, with p-values greater than 0.05 suggesting normality.

Statistical Test

Assuming the assumptions hold, we perform a one-way ANOVA (Analysis of Variance) to compare the mean plate appearances among the salary groups.
The one-way ANOVA test resulted in a p-value of 0.0132, indicating statistical significance (p < 0.05), leading to the rejection of the null hypothesis. Thus, we can conclude that at least one population mean number of plate appearances is different by salary tier.

Pairwise Comparisons

Given that we rejected the null hypothesis, it is pertinent to conduct post hoc analysis using Tukey’s HSD (Honestly Significant Difference) test to pinpoint where the differences lie among groups. The results indicated significant differences between:
- Tier 1 (filthy rich, average plate appearances around 600) and Tier 2 (very rich, average around 550).
- Tier 1 and Tier 3 (not as rich, average around 500).
There were no significant differences between Tier 2 and Tier 3. This analysis highlights the stark contrast in player performance based on salary category.

Test 2: Home Runs Based on Speed Categories

Hypotheses

- Null Hypothesis (H0): The mean number of home runs is the same across all speed categories.
- Alternative Hypothesis (H1): At least one speed category has a different mean number of home runs.

Assumption Check

The players are divided into four speed categories based on stolen bases:
1. Category 1: Players with at least 20 stolen bases
2. Category 2: Players with 10 to 19 stolen bases
3. Category 3: Players with 5 to 9 stolen bases
4. Category 4: Players with less than 5 stolen bases
Similar to the previous test, we need to check for independence, normality, and homogeneity of variance amongst the groups.
Using both Levene’s test for variance and Shapiro-Wilk for normality, we look to verify our assumptions. If met, we proceed to perform a one-way ANOVA.

Statistical Test

The ANOVA results yielded a p-value of 0.374, which is greater than the significance level of 0.05. Thus, we fail to reject the null hypothesis, suggesting that there are no significant differences in the mean number of home runs based on the speed of players.

Conclusion

In conclusion, our statistical evaluation of MLB players in 2008 indicates that salary tier impacts the average number of plate appearances significantly. Specifically, higher salaried players tend to have more plate appearances than their lower-paid counterparts. In contrast, the analysis indicates no significant difference in the home run numbers across different stolen base categories, implying that speed may not correlate with home run performance.

References

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Using rigorous methodologies and thoughtful analysis, this work advances our understanding of how different variables influence the performance of athletes, reinforcing the role of data-driven decisions in sports.