Question

How does Alex Smith, quarterback for the Kansas City Chiefs, compare to Joe Flacco, quarterback for the Baltimore Ravens? The following table shows the number of completed passes for each athlete during the 2017 NFL football season: $$ \begin{array}{ccc|ccc} \hline \multicolumn{3}{c|} {\text { Alex Smith }} & \multicolumn{3}{|c} {\text { Joe Flacco }} \\ \hline 25 & 27 & 29 & 25 & 22 & 19 \\ 23 & 25 & 27 & 29 & 34 & 31 \\ 20 & 14 & 16 & 26 & 10 & 8 \\ 19 & 25 & 21 & 20 & 27 & 25 \\ 23 & 19 & 28 & 23 & 24 & 9 \\ & & & 20 & & \\ \hline \end{array} $$ Use the Wilcoxon rank sum test to analyze the data and test to see whether the population distributions for the number of completed passes differ for the two quarterbacks. Use \(\alpha=.05 .\)

Short answer

Based on the Wilcoxon Rank Sum test performed with a significance level of \(\alpha = 0.05\), we can conclude that there is a significant difference between the population distributions for the number of completed passes for Alex Smith and Joe Flacco during the 2017 football season.

Step-by-step solution

1. Combine and Rank Observations

First, combine all the completed passes from both Alex Smith and Joe Flacco into a single list and rank them. Be mindful that not all table cells have data, ignore the empty ones: $$ \{19,14,16,19,25,21,20,23,25,27,23,19,28,29,27,29,25,25,22,19,26,20,27,25,23,24,20,29,34,31,10,8,23,9\} $$ Now, rank these values from lowest to highest, assigning the average rank to any tied values: $$ \{1,2,3,4,5.5,7.5,8,10,11,12.5,14.5,16.5,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37\} $$

2. Calculate Sum of Ranks

Next, find the sum of the ranks for each quarterback. Separate the ranks for Alex Smith and Joe Flacco in two sets: - Alex Smith: {1, 2, 3, 4, 5.5, 7.5, 8, 10, 11, 12.5, 14.5, 16.5, 17, 18, 19, 20, 21, 22} - Joe Flacco: {23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37} Calculate the sum of the ranks for each quarterback: - \(R_A = 254.5\) - \(R_J = 528.5\)

3. Calculate Test Statistic

Now, we need to calculate the test statistic, W. Since the sample size of Alex Smith's data is smaller than Joe Flacco's, we'll use \(R_A\) to calculate W: $$ W = R_A - \frac{n_A(n_A + 1)}{2} $$ Where \(n_A\) is the number of observations in Alex Smith's sample. In this case, \(n_A = 18\), so the calculation becomes: $$ W = 254.5 - \frac{18 (18 + 1)}{2} = 254.5 - 153 = 101.5 $$

4. Determine Critical Value

As we are given \(\alpha = 0.05\), we can look up the critical value, \(W_{crit}\), in a Wilcoxon Rank Sum test table, or use a software or online tool, for the sample sizes \(n_A=18\) and \(n_J=15\). In this case, we find that \(W_{crit} = 130\).

5. Compare Test Statistic and Critical Value

Now we need to compare \(W\) and \(W_{crit}\) to see if there is a significant difference between the two quarterbacks' completed passe distributions: - If \(W > W_{crit}\), there is not a significant difference between the distributions. - If \(W \leq W_{crit}\), there is a significant difference between the distributions. As we found \(W = 101.5\) and \(W_{crit} = 130\), this means: $$ 101.5 \leq 130 $$ So there is a significant difference between the population distributions for the number of completed passes for Alex Smith and Joe Flacco at a significance level of \(\alpha = 0.05\).

Key concepts

Non-parametric Statistical Tests

Non-parametric statistical tests are a key tool in the analysis of data that does not assume a specific distribution for the population from which samples are drawn. Unlike their parametric counterparts, non-parametric methods do not require the assumption of normality. They are used when data is ordinal, when ranks are more meaningful than raw data, or when the sample size is very small, making it difficult to meet the assumptions of parametric tests.

One well-known non-parametric test is the Wilcoxon rank sum test, which is used for comparing two independent samples to determine if they come from the same distribution. This test is particularly useful when dealing with non-normal data or when the data's scale does not have the interval properties that permit the use of parametric tests like the t-test. In the exercise, the Wilcoxon rank sum test was applied to compare the number of completed passes for quarterbacks Alex Smith and Joe Flacco during the NFL season.

Probability and Statistics

Probability and statistics are branches of mathematics concerned with the laws governing random events and the collection, analysis, interpretation, and presentation of numerical data, respectively. Understanding them is essential for performing hypothesis testing and using non-parametric statistical tests effectively.

The Wilcoxon rank sum test is deeply grounded in statistical theory and principles of probability. Probability is utilized to describe the likelihood of a particular outcome occurring under the null hypothesis, while statistics help us describe and infer from our collected data. In the exercise, the outcomes of completed passes could be influenced by many unpredictable factors, which makes probability an essential concept in interpreting these outcomes statistically.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It typically starts with a null hypothesis (\( H_0 \)), which states that there is no effect or difference between groups or conditions, and an alternative hypothesis (\( H_A \text{ or } H_1 \text{) that signifies a significant effect or difference.

In the context of the Wilcoxon rank sum test, the null hypothesis could be that there is no difference between the distribution of completed passes for Alex Smith and Joe Flacco. The alternative hypothesis would be that there is a difference. We compare the test statistic, in this case the sum of ranks for one group, to a critical value that corresponds to the desired level of significance, typically \text{0.05, to make a decision about whether to reject the null hypothesis. If the test statistic falls below this critical value, the null hypothesis is rejected, indicating a statistically significant difference. In the exercise, the conclusion was that there was a significant difference between the quarterbacks' performance at a significance level of \text{0.05, leading to the rejection of the null hypothesis.}}