91Ó°ÊÓ

The following table gives the 2015 total payroll (in millions of dollars) and the percentage of games won during the 2015 season by each of the National League baseball teams. $$ \begin{array}{lcc} \hline \text { Team } & \begin{array}{c} \text { Total Payroll } \\ \text { (millions of dollars) } \end{array} & \begin{array}{c} \text { Percentage of } \\ \text { Games Won } \end{array} \\ \hline \text { Arizona Diamondbacks } & 92 & 49 \\ \text { Atlanta Braves } & 98 & 41 \\ \text { Chicago Cubs } & 119 & 60 \\ \text { Cincinnati Reds } & 117 & 40 \\ \text { Colorado Rockies } & 102 & 42 \\ \text { Los Angeles Dodgers } & 273 & 57 \\ \text { Miami Marlins } & 68 & 44 \\ \text { Milwaukee Brewers } & 105 & 42 \\ \text { New York Mets } & 101 & 56 \text { Philadelphia Phillies } & 136 & 39 \\ \text { Pittsburgh Pirates } & 88 & 61 \\ \text { San Diego Padres } & 101 & 46 \\ \text { San Francisco Giants } & 173 & 52 \\ \text { St. Louis Cardinals } & 121 & 62 \\ \text { Washington Nationals } & 165 & 51 \\ \hline \end{array} $$ Compute the linear correlation coefficient, \(\rho .\) Does it make sense to make a confidence interval and to test a hypothesis about \(\rho\) here? Explain.

Step by step solution

01

Calculate the linear correlation coefficient

First, we need to calculate the correlation coefficient \(r\) between the total payroll (in millions of dollars) and the percentage of games won. We use the following formula to calculate \(r\): \[ r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}}\] Where \(x\) is the total payroll, \(y\) is the percentage of games won, \(n\) is the number of teams (in this case, 15), \(\Sigma xy\) is the sum of the product of corresponding \(x\) and \(y\) values, \(\Sigma x\) and \(\Sigma y\) are the sum of \(x\) and \(y\) values respectively, \(\Sigma x^2\) and \(\Sigma y^2\) are the sums of the squares of \(x\) and \(y\) values respectively.

02

Decide whether it makes sense to construct a confidence interval for \(\rho\)

After calculating \(r\), the sample correlation coefficient, we need to consider whether it makes sense to construct a confidence interval for the population correlation coefficient \(\rho\). Generally, it would make sense if we have reason to believe that our sample (the teams in the given year) is representative of a larger population (all baseball teams in different years), and if we assume that the relationship between payroll and percentage of games won is linear in this population. This is more of an interpretive step and thus might differ depending on individual reasoning.

03

Decide whether it makes sense to test a hypothesis about \(\rho\)

Whether it makes sense to test a hypothesis about \(\rho\) would largely depend on the same considerations as above - if we believe our sample is representative and the relationship between payroll and games won is linear in the population. If \(\rho\) is close to 0, it might not make sense to test a hypothesis (like \(\rho \neq 0\)), since a result very close to 0 would mean there's little linear relationship. However, if \(\rho\) is significantly different from 0, then testing for \(\rho \neq 0\) could make sense, as there could be potential evidence of a linear relationship.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation Analysis

Correlation analysis is a statistical method used to measure the strength and direction of the linear relationship between two variables. In this exercise, we are examining the relationship between the total payroll of each team and the percentage of games they won. This involves calculating the linear correlation coefficient, often denoted as \(r\).

The correlation coefficient \(r\) can range from -1 to 1. A value of 1 implies a perfect positive linear relationship, where as the payroll increases, the percentage of games won does too. Conversely, a value of -1 indicates a perfect negative linear relationship, meaning that as the payroll increases, the percentage of games won decreases. A value close to 0 signifies no linear relationship between the variables.

• Positive correlation: As one variable increases, the other also increases.
• Negative correlation: As one variable increases, the other decreases.

By computing \(r\), we can quantify how payroll is related to team performance in terms of games won. This analysis is crucial for teams to understand the impact of financial strategies on sporting success.

Hypothesis Testing

Hypothesis testing is a formal process used to make inferences or draw conclusions about a population based on sample data. In the context of this exercise, hypothesis testing would involve making claims about the population correlation coefficient \(\rho\).

Typically, we might test the null hypothesis \(H_0: \rho = 0\), which states there is no linear relationship between the variables in the population. The alternative hypothesis \(H_a: \rho eq 0\) suggests there is a linear relationship.

• If the test statistic calculated from the sample data results in a p-value less than the chosen significance level (usually 0.05), we reject \(H_0\) in favor of \(H_a\).
• This implies there's sufficient evidence to suggest a linear relationship exists in the population.

Before conducting hypothesis testing, it's important to ensure that the sample data is representative of the population. This supports the validity of any inferences made about the larger baseball team population.

Confidence Interval

A confidence interval provides a range of values that is likely to contain the population correlation coefficient \(\rho\). This interval is associated with a confidence level, often 95%, which suggests that we are 95% confident the true \(\rho\) lies within this range.

Creating a confidence interval for \(\rho\) after computing the sample correlation coefficient \(r\) is helpful when we assume our sample is indicative of the broader population of baseball teams.

• The width of the confidence interval represents the precision of our estimation, influenced by sample size and variability.
• A narrower interval indicates more precision, while a wider interval suggests greater uncertainty.

When the confidence interval does not include 0, it provides additional evidence supporting a significant linear relationship between payroll and game performance.

Linear Relationship

A linear relationship exists when two variables change in a consistent manner, represented by a straight line on a graph. In correlation analysis, establishing a linear relationship between payroll and games won means predicting the percentage of games won based on payroll should follow a linear trend.

• If the scatter plot of (payroll, games won) points creates a visual pattern that can be approximated with a straight line, we say there is a linear relationship.
• Understanding this relationship helps in making strategic decisions regarding team management and financial allocations.

While the linear correlation coefficient reveals the strength of this relationship, it does not point out cause and effect. External factors could also influence both payroll and team success, making it crucial to contextualize the findings.