15A Coreq

In the next preview assignment and in the next class, you will need to be able to calculate probabilities and expected counts for one categorical variable. Probability & Expected Value ofOne Categorical VariableIn previous in-class activities, we explored probability and expected counts in various contexts. We will build upon this understanding to get expected counts associated with the categories of a categorical variable. Italian Soccer: Italian youth soccer leagues create cohorts of children based on year of birth. For example, children born in 2005 only played other children born in that same year. If a child was born on December 31,2004, they played with the 2004 cohort (rather than the younger 2005 cohort). So, children born earlier in the year (e.g., January or February) tend to be the eldest players in their leagues. Children born later in the year (e.g.,November or December) tend to be the youngest players in their leagues.Could this seemingly unimportant practice—grouping by year of birth—have an effect on players’ later soccer careers? We’re going to explore this question in this corequisite support activity and the next preview assignment.

Question 1

1) A calendar year can be defined in quarters (as shown in the following table). Let’s assume birth rates are the same across all quarters and that quarters are equally long. Fill in the probability table.

Quarter Quarter 1

(Jan. – March)

 Quarter 2

(April – June)

 Quarter 3

(July – Sept.)

 Quarter 4

(Oct. – Dec.)
Probability of being born this quarter

Question 2

2) Assume you collected a random sample of 200 Italian soccer players who played in youth leagues and then went on to play professional soccer.
Part A: Making the same assumptions as before (equal birth rates across quarters), how many of these soccer players would youexpect to have birthdates in each quarter? Fill in the following tableand show your calculations.
Quarter Quarter 1

(Jan. – March)

 Quarter 2

(April – June)

 Quarter 3

(July – Sept.)

 Quarter 4

(Oct. – Dec.)
Expected number of soccer players born this quarter
Part B: Interpret the number you found for Quarter 1. What does that number mean?
Part C: Was the expected number the same or different between quarters? Explain.

Question 3

3)In reality, birth rates are not the same between quarters of the year. Some quarters are longer (have more days), and different cultures have different preferences for times of birth. Researchers measured birth rates in Italy in the years in which professional soccer players were born and found the following results:

Quarter Quarter 1

(Jan. – March)

 Quarter 2

(April – June)

 Quarter 3

(July – Sept.)

 Quarter 4

(Oct. – Dec.)
Proportion of births in Italy 0.2248 0.2498 0.2574 0.2680

Part A: Among a random sample of Italians, which quarter would you expect to have the highest number of players? Which quarter would you expect to have the lowest number of players?Explain.Part B: Imagine a researcher selecteda random sample of eightItalians. Among them, there were more Quarter 1 birthdatesthan Quarter 4 birthdates. Is this possible?Is it likely?Explain.Part C: Imagine a researcher selecteda random sample of 800,000 Italians. Among them, there were more Quarter 1 birthdatesthan Quarter 4 birthdates. Is this possible? Is it likely?Explain.

Fumarco, L.& Rossi, G. (2018, August 8).The relative age effect on labour market outcomes -Evidence from Italian football. EuropeanSport Management Quarterly,18(4), 501–516. DOI:10.1080/16184742.2018.1424225

Question 4

4)Let’s turn back to our problem. Remember that you collected data on 200professional Italian soccer players. Part A: Use the true birth rates in Italy to find the expected counts for each quarter, assuming the players’birthdates are distributed similarly to the birthdates of the general Italian population. When filling in the following table, show your calculations.
Quarter Quarter 1

(Jan. – March)

 Quarter 2

(April – June)

 Quarter 3

(July – Sept.)

 Quarter 4

(Oct. – Dec.)
Expected number of soccer players born this quarter  

 

  

 

 

  

 

  

 
Part B:What trend do you notice in these expected counts? How do they shift from the beginning of the year to the end?

Question 5

5)The observed birthdates of an actual sample2 of 1,703 professional Italian soccer players aredisplayed in the following table.

Quarter Quarter 1

(Jan. – March)

 Quarter 2

(April – June)

 Quarter 3

(July – Sept.)

 Quarter 4

(Oct. – Dec.)
Observed number of soccer players 507 534 389 273
Part A: Do these observed birthdates have a different pattern than the expected birthdates? If so, describe the difference.2Fumarco, L. & Rossi, G. (2018, August 8). The relative age effect on labour market outcomes -Evidence from Italian football. EuropeanSport Management Quarterly, 18(4), 501–516. DOI:10.1080/16184742.2018.1424225
Part B:Do you believe the differences between the observed and expected countsoccurred by chance alone? Or, do you believe that professional Italian soccer players tend to truly have different patterns in birthdate distribution than the general population? Explain.
2Fumarco, L. & Rossi, G. (2018, August 8). The relative age effect on labour market outcomes -Evidence from Italian football. EuropeanSport Management Quarterly, 18(4), 501–516. DOI:10.1080/16184742.2018.1424225