University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

232

Pages:

3
Author:

Niklas

Exercise 1. Suppose you are requested to describe a qualitative variable: a. Which central tendency and variability measures would be appropriate? When we are dealing with qualitative variables and we want to summarize the distribution with central tendency measures, we can only use the mode of the distribution. Instead, to quantify how different the modalities of statistical units are, we can compute mutability indexes such as the Gini index and the Shannon entropy. b. Give an interpretation of these measures. The mode represents the highest frequency of occurrence in the data and, indeed, it can be computed also for qualitative variables as opposed to the median and the mean. Gini index and Shannon entropy attempt to quantify heterogeneity-homogeneity of units with respect to a given variable. Both indexes go to zero when mutability is minimum and assume maximum value when mutability is maximum. Exercise 2 a. Complete the table: Subscription North Center South Total Sky 25.2 18 16.8 60 Netflix 18.8 12 9.2 40 Total 44 30 26 100 b. What is the most present company in Northern Italy? To determine the most present company in Northern Italy, we calculate the percentages of Sky and Netflix subscribers in the North region: P(Sky | North) = (25.2 / 44) * 100 = 57.3% P(Netflix | North) = (18.8 / 44) * 100 = 42.7% Therefore, Sky is the most present company in Northern Italy. c. Is there association between the two variables? Try to answer looking at the conditional distributions, then compute a proper index to verify your guess. To determine if there is an association between the two variables, we can compare the conditional distributions: Conditional distribution of Sky given the region: North: 25.2% Center: 60% South: 64.6% Conditional distribution of Netflix given the region: North: 42.7% Center: 40% South: 35.4% From these distributions, it appears that there might be a weak association between the two variables. Sky is more popular in the South, while Netflix is more popular in the North. To verify this guess, we can compute the chi-square statistic: χ² = Σ [(O - E)² / E] where O is the observed frequency and E is the expected frequency. After calculating the chi-square statistic and comparing it to the critical value from the chi-square distribution, we can determine if there is a significant association between the two variables. Exercise 2 (continued) c. Is there association between the two variables? By looking at the conditional distributions: P(Sky | North) = 57.3% P(Netflix | North) = 42.7% P(Sky | Center) = 60% P(Netflix | Center) = 40% P(Sky | South) = 64.6% P(Netflix | South) = 35.4% We can say that there is not a high variability. Now we calculate the variability through the Pearson's chi-square index (χ²): χ² = Σ Σ [(nij - nᵢ•nj•) / nᵢ•nj•]² where nij is the observed frequency, nᵢ• is the row total, and nj• is the column total. After calculating the chi-square statistic, we find that it is not zero, indicating that there is some degree of association between the two variables. However, the value of χ² is not very large, suggesting that the association is not very strong.