Chi-Squared Goodness of Fit Test

• One categorical variable with counts (or proportion) in each category

• We have seen: products are produced by two machines, machine A produced 15 defective parts in a run of 280, while machine B produced 10 defective parts in a run of 200. Is there a difference in the reliability of these two machines?

• New question type: products are produced by three machines, machine A produced 15 defective parts in a run of 280, while machine B produced 10 defective parts in a run of 200. Is there a difference in the reliability of these machines?

Practice Question 1

In the past, for a large introductory statistics course, the proportions of students that received grades of A, B, C, D, or F have been, respectively, 0.15, 0.35, 0.30, 0.10, and 0.10

This year, there were 200 students in the class, and following grades were given:

Test to see whether the distribution of grades this year was different from the distribution in the past?

• Hypothesis

  • H0: PA = 0.15, PB = 0.35, PC = 0.30, PD = 0.10, PF = 0.10

  • H1: at least one p does not fit the distribution

• Calculate expected values

• Conditions

  • Random

  • Independent

  • Count: At least 80% of the expected counts are greater than 5 and none are less than 1

• Calculate

  

  

  

• Calculate by calculator

  

  

• P-value

  df = k - 1 (k: number of categories)

  

• Interpret

  • P < α

  • So we reject the null hypothesis and have evidence to support the claim that at least one grade proportion does not fit the expected distribution

Chi-Squared Test of Homogeneity or Independence/Association

• Two categorical variables

• Homogeneity

  • Do two or more sub-groups of a population share the same distribution of a categorical variable (each group has its own sample)

  • Do people of different races have the same proportion of smokers to non-smokers.

  • Do different education levels have different proportions of Democrats, Republicans, and Independent

• Independence/Association

  • Determining whether two categorical variables are associated (variables from a single SRS)

  • Is there an association between race and smoking status

  • Is there an association between education and voting preference

Practice Question 2

Girls and boys at an elementary school were sampled and asked about their favorite subject

1. Does favorite subject differ by gender?

• Hypothesis

  • H0: favorite subject does not differ by gender

  • H1: favorite subject does differ by gender

• Expected

  • Row Total * Colum Total / Total

• Conditions

  • For each sub group, the sample is a SRS

  • NO expected cell counts are < 5

• Calculate

  

• Calculate by calculator

  

  

  

• P-value

  • df = (r-1)*(c-1)

• Interpret

  • P < α

  • So we reject the null hypothesis and have evidence to support the claim that favourite subject is different between boys and girls

1. Is favourite subject associated with gender?

• H0: There is no association between favorite subject and gender

• H1: There is an association between favorite subject and gender

Practice Question 3

You are playing a dice game with a friend. They brought a 6 sided die that you think may not be fair. You conduct an experiment to determine if it is fair. You roll the die 100 times and get following:

• Hypothesis

  • H0: P1 = P2 = P3 = P4 = P5 = P6

  • H1: at least one is not equal

• Expected

  • 1/6 = 0.1667

• Conditions

  • Random

  • Independent

  • Expected counts are greater than 5

• Calculate

  

• Calculate by calculator

  

• Interpret

  • P > α

  • So we fail to reject the null hypothesis and do not have evidence to support the claim that the die is unfair