Lecture 19 - Contingency Tables and Two Sample Tests of Proportions

Housekeeping

• Today’s plan 📋

  • Comments and Questions about Previous Lecture from Engangement Questions

  • Upcoming Dates

  • A few minutes for R Questions 🪄

  • Review Question - Two-sided test

  • Review of One Sample Proportion Hypothesis Tests

  • Contingency Tables

    • Tests of Two proportions

    • Format of Hypothesis Tests

Review: R and RStudio 🪄

• Review: You have two options to facilitate your introduction to R and RStudio:

  • Option 1: Create Posit Cloud account and download and install R and RStudio on your laptop.
  • Option 2: Start with free Posit Cloud account and use that and later transition to using R/Rstudio on your laptop.

• If you are comfortable with coding: Start with Option 1, but still sign up for Posit Cloud account.

  • We will use Posit Cloud for Quizzes.

• If you are nervous about coding: Choose Option 2.

• For both options: I can help with download/install issues during office hours.

• What I do: I maintain a Posit Cloud account for helping students but I do most of my work on my laptop.

• NOTE: We will use R and RStudio in class during MOST lectures

  • You can use either Posit Cloud or your laptop.

Upcoming Dates

• HW 6 is due 11/1 (Grace period ends 11/3)

  • Demo videos are posted on Blackboard

  • This assignment seems long but it’s not.

  • It consists of just three hypothesis tests with questions about each test.

  • Most questions are multiple choice, but do not just guess and keep trying.

• HW 7 will be posted on Wednesday (Lectures 18 - 20)

• Test 2 is on November 14th and will include material up through Lecture 20

  • Lecture 21 - Intro to Portfolio Management will be on Final Exam, not on Test 2.

💥 Lecture 19 In-class Exercises - Q1 (Review) 💥

Review of One Sample Hypothesis Tests of Proportions

Should We End Daylight Savings Time?

YouGov Polled 987 US adults

• 612 said YES, we should eliminate the practice of changing our clocks.

• 375 said NO or they were unsure. We group these two categories together.

• \(H_{0}: P_{YES} = P_{NO}\) - No difference in proportion with these two opinions.
• \(H_{A}: P_{YES} \neq P_{NO}\) - There is a difference in proportion with these two opinions.

Should We End Daylight Savings Clock Changes?

prop.test(612,987,correct=F)

    1-sample proportions test without continuity correction

data:  612 out of 987, null probability 0.5
X-squared = 56.909, df = 1, p-value = 0.00000000000004565
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
 0.5893699 0.6498207
sample estimates:
        p 
0.6200608 

One Sample Hypothesis Test Conclusion

Hypotheses being tested:

• \(H_{0}: P_{YES} = 0.5\)

  • The Yes and Not Yes votes are roughly equal.

• \(H_{A}: P_{YES} \neq 0.5\)

  • There is a difference in the proportion Yes and Not Yes votes.

• P-value from hypotheses test: < 0.0001

• Conclusion: P-value is much less than 0.05 so we REJECT \(H_{0}\).

• Interpretation: See Polling Question on next slide

💥 Lecture 19 In-class Exercises - Q2 💥

Given our stated hypotheses and our p-value < 0.0001

• \(H_{0}: P_{YES} = 0.5\)

• \(H_{A}: P_{YES} \neq 0.5\)

Contingency Tables to Examine Proportions in Multiple Categories

Comparing Contingency Tables and Plots

Contingency tables and bar plots are two effective ways to examine these data

Hypothesis Test Comparing Two Proportions

• Hypotheses being tested:

  • \(H_{0}: P_{18-29} = P_{30-44}\)
    • There is no difference between these two age groups with respect to proportion that says yes.
  • \(H_{A}: P_{18-29} \neq P_{30-44}\)
    • There is a difference between these two age groups with respect to proportion that says yes.

x <- c(99, 129)  # yes votes in each age group (18-29 first)
n <- c(197, 234) # sample size in each age group (18-29 first)

prop.test(x,n, correct=F)

    2-sample test for equality of proportions without continuity correction

data:  x out of n
X-squared = 1.0199, df = 1, p-value = 0.3125
alternative hypothesis: two.sided
95 percent confidence interval:
 -0.14327319  0.04578523
sample estimates:
   prop 1    prop 2 
0.5025381 0.5512821 

Two Proportion Hypothesis Test Conclusion and Interpretation

• Hypotheses being tested:

  • \(H_{0}: P_{18-29} = P_{30-44}\)
    • No difference between these age groups respect to proportion that says yes.
  • \(H_{A}: P_{18-29} \neq P_{30-44}\)
    • There is a difference between these age groups respect to proportion that says yes.

💥 Lecture 19 In-class Exercises - Q3 and Q4 💥

Question 3: What is the p-value from this hypothesis test?

Question 4: If we specify \(\alpha = 0.05\), do we reject or fail to reject the null hypothesis, \(H_{0}\)?

💥 Lecture 19 In-class Exercises - Q5 and Q6 💥

Question 5: What do we conclude about the opinions of these two age groups?

Question 6 (Not on Point Solutions): What type of error might we have made?

Do Gen-Zs and Millenials differ from Gen-Xers?

Column and Row Percentages

Hypothesis Test Comparing Two Proportions

• Hypotheses being tested:

  • \(H_{0}: P_{18-44} = P_{45-64}\)
    • No difference between these age groups respect to proportion that says yes.
  • \(H_{A}: P_{18-44} \neq P_{45-64}\)
    • There is a difference between these age groups respect to proportion that says yes.

x <- c(228,201)  # yes votes in each age group (18-44 first)
n <- c(433,319)  # sample size in each age group (18-44 first)

prop.test(x,n, correct=F)

    2-sample test for equality of proportions without continuity correction

data:  x out of n
X-squared = 8.0355, df = 1, p-value = 0.004587
alternative hypothesis: two.sided
95 percent confidence interval:
 -0.17437592 -0.03269438
sample estimates:
   prop 1    prop 2 
0.5265589 0.6300940 

💥 Lecture 19 In-class Exercises - Q7 💥

What do you conclude from this two sample two-sided hypothesis test of two proprtions?

Key Points from Today

• Protocol for conducting and interpreting hypothesis tests is same, regardless of how they are specified.

• This is true for quantitative data and for categorical proportion data

• For two sample tests of proportions, it is helpful to examine the data using contingency tables.

• By default, it is common for two sample tests of proportions to be conducted as two sided tests.

• These same methods can be used with larger contingency tables tat are interatively analyzed.