Preparing for the next classIn the next in-class activity, you will need to be able to calculate the value of a chi-square test statistic and explain what it means in the context of a claim. You will also need to write the null and alternative hypotheses for a chi-square test for goodness of fit. We will now revisit the Italian soccer scenario. Recall that researchers measured birth rates in Italy and found the following results:
QuarterQuarter 1(Jan. –March)Quarter 2 (April –June)Quarter 3(July –Sept.)Quarter 4(Oct. –Dec.)Proportion of births in Italy22.48%24.98%25.74%26.80%
Question 1
1) Imagine we wanted to know if the distribution of birthdate quarters among Italian soccer players was truly different from that of the general Italian population. How could we formulate this question as two testable hypotheses?
Imagine that three different teams of researchers gathered birthdate data on three different samples of professional Italian soccer players. The following tables(continued on the next page)show their results.Sample AQuarter 1(Jan. –March)Quarter 2 (April –June)Quarter 3(July –Sept.)Quarter 4(Oct. –Dec.)Observed number of soccer players3421Sample BQuarter 1(Jan. –March)Quarter 2 (April –June)Quarter 3(July –Sept.)Quarter 4(Oct. –Dec.)Observed number of soccer players3,0004,0002,0001,000 Sample CQuarter 1(Jan. –March)Quarter 2 (April –June)Quarter 3(July –Sept.)Quarter 4(Oct. –Dec.)Observed number of soccer players507534389273
The data in the tables are displayed in the following side-by-side barchart:
Question 2
2) Do you believe that professional Italian soccer players’ birthdate distribution differs from that of the general Italian population? Rank the results in order from most convincing to least convincing and explain.Hint: Think both about the sample size (total number of players) and the imbalance between categories.
Question 3
3) Chi-square statistic valueswere calculated foreach of these samples (usingexpected counts under the general Italian population distribution). These statistics are displayed in the following table.Note: Sample A had too few players to meet the conditions of a chi-square test, but we will discuss theseissues during the in-class activity.Sample ASample BSample CChi-Square2.342,336147
Part A: Give an intuitive explanation for why Sample B had the highest value and why Sample A had the lowest value.
Part B: Does a greater chi-square test statistic indicate more convincing or less convincing evidence that the distribution of birthdates is different for Italian soccer players than in the general population? Explain.
Question 4
4) Sample C was theactual sample obtained by researchers in the study mentioned previously.Does this sample provide enough evidence to reject the null hypothesis and support the alternativehypothesis? Let’s investigate.
Part A: Assuming certain conditions are met (we will talk about these conditions inthe in-class activity), we can compare our chi-square test statistic value to the chi-square distribution to get the probability of obtaining the distribution we observed or one thatdiffersmore from our expectations by chance aloneif the distributionof soccer players is the sameas the general Italian population. Let’s do this for the chi-square statistic from Sample C.Goto theDCMPChi-Square Distributiontool at https://dcmathpathways.shinyapps.io/ChisqDist/.Click the Find Probabilitytab at the top of the data analysis tool. Choose the appropriate degrees of freedom (number of categories -1) and select the “Upper Tail” probability type. Enter the calculated chi-square statistic and write downthe probability displayed in the data analysis tool to the nearest whole number.
Part B: Would you say that Sample Cprovides convincingevidence that the distribution of birthdates among Italian soccer players is different from the distribution among the general Italian population? Explain.
Part C: Expand upon your conclusion. If you found convincing evidence that the distribution of birthdates among Italian soccer players is different than that of the general Italian population, provide one possible explanation for this phenomenon. If you didn’t find convincing evidence of a difference, provide one possible explanation forthis outcome.
