Lecture 18 - Two Sample t-Tests and Confidence Intervals

Housekeeping

• Today’s plan 📋

  • Comments and Questions about Previous Lecture from Engangement Questions

  • Upcoming Dates

  • A few minutes for R Questions 🪄

  • Review Question - One sample Hypothesis Test

  • One-sided vs Two-sided Tests

  • Two Sample Hypothesis tests

    • t-test comparing Two Independent groups

    • Paired t-tests

  • Introduction of HW 6

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

• Mid-Semester Progress Reports will be submitted this week

  • Will not include attendance and in-class polling

  • Not everyone receives a report (no news is medium to good news).

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

  • 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 next week (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 18 In-class Exercises - Q1 (Review) 💥

Review of Types of One-Sided Hypothesis Tests

Reminder of How to Interpret P-values

• In a standard situation when we use \(\alpha=0.05\), I think of the evidence against the null hypothesis along this spectrum:

  • 0.0 - 0.01 Very strong evidence against \(H_{0}\)

  • 0.011 - 0.03 Strong evidence against \(H_{0}\)

  • 0.031 - 0.049 Some evidence against \(H_{0}\)

  • 0.05 - 0.07 Suggestive evidence against \(H_{0}\)

  • 0.071 - 0.099 Minimal evidence against \(H_{0}\)

  • 0.1 and above No evidence against \(H_{0}\)

Hypothesis Test Framework for all Hypothesis Tests

• Hypothesis Tests can be structured differently BUT framework for testing is the same.

  1. Specify hypotheses that match data and question of interest.

  2. Specify \(\alpha\).

  3. Conduct hypothesis test using software.

     • e.g. t.test and prop.test commands in R

  4. Compare p-value to \(\alpha\) and/or examine confidence interval or confidence bounds.

     • P-value \(\lt \alpha\): Reject \(H_{0}\).

     • P-value \(\geq \alpha\): Do not reject \(H_{0}\).

  5. Interpret results in terms of data and question asked.

     • What do we conclude and what type of error might we have made?

Two sample Hypothesis Test Example

Question: Is there a significant difference between the city miles per gallon (MPG) of Italian and German high end cars?

1. Setting up hypotheses

• Question as written assumes no prior knowledge.

• If we don’t know which group mean might be larger, we use a two-sided test.

• Null (default) hypothesis is that the two group means are equal

• Alternative hypothesis is the complement, the two group means are unequal

• Hypotheses:

  • \(H_{0}: \mu_{ITA} = \mu_{GER}\)

  • \(H_{A}: \mu_{ITA} \neq \mu_{GER}\)

2. Specifying \(\alpha\), the cutoff value. We use \(\alpha = 0.05\) unless we have a reason not to.

Two sample Hypothesis Test Example

Question: Is there a significant difference between the city fuel efficiency of Italian and German high end cars?

3. Conduct the t.test to answer this question

   • Note that in t.test command below alternative="two.sided" and conf.level=.95 are the defaults and not required.
   • The defaults are included here so you will no how to change them.

💥 Lecture 18 In-class Exercises - Q2 and Q3 💥

4. Compare p-value to \(\alpha\) and/or examine confidence interval or confidence bounds.

• Use the output from the t-test to answer these questions:

💥 Lecture 18 In-class Exercises - Q4 and Q5 💥

5. Interpret results in terms of data and question asked.

Relationship between One-sided and Two-sided Tests

The P-value from our original two-sided test is 0.0004092

Lets rerun the test above as a left-sided test and examine the change in the results.

t.test(itly_cty, gmny_cty, 
       alternative="less", conf.level=.95) 

    Welch Two Sample t-test

data:  itly_cty and gmny_cty
t = -4.1147, df = 23.461, p-value = 0.0002046
alternative hypothesis: true difference in means is less than 0
95 percent confidence interval:
      -Inf -2.612611
sample estimates:
mean of x mean of y 
   13.400    17.875 

• One-sided p-value = \(\frac{1}{2}\) of two-sided p-value.

• If we know we are only interested in one tail, p-value is cut in half

• Some disciplines ONLY use two-sided tests so that rejecting \(H_{0}\) is more difficult.

• Other disciplines ONLY use one-sided tests, because direction of the test is obvious.

A Paired t-test Example

• It is fairly well known that city gas mileage is ALWAYS lower than highway mileage.

• We can verify that this is true for these European cars using a paired t-test.

• In a paired t-test, you look at the same individuals (cars, people, sites, etc) under two sets of conditions.

• A common use of paired t-tests is two collect data on humans before and after some activity such as a scary movie or period of exercise.

• The key criteria for using a paired t-test is that two sets of data are NOT from independent groups.

• A paired t-test is a ONE SAMPLE T-TEST of the DIFFERENCES

Comparing City and Highway MPG

Do these European cars get better mileage on the highway?

1. Specify hypotheses for a right-sided test (Hypotheses have same format as two sample t-tests):

   • \(H_{0}: \mu_{HWY} \leq \mu_{CTY}\)

   • \(H_{0}: \mu_{HWY} \gt \mu_{CTY}\)

2. Specify \(\alpha = 0.05\)

3. Conduct paired t-test of these data:

cty <- euro_cars$mpg_c    # save variables to short names
hwy <- euro_cars$mpg_h
t.test(hwy, cty, alternative="greater", conf.level=.95, paired=T) 

    Paired t-test

data:  hwy and cty
t = 16.913, df = 30, p-value < 0.00000000000000022
alternative hypothesis: true mean difference is greater than 0
95 percent confidence interval:
 6.181461      Inf
sample estimates:
mean difference 
       6.870968 

A Closer Look at the Paired t-test

• A paired t-test is ONLY appropriate when the same individuals are measures twice under different conditions.

• In order to do a paired t-test, the software creates a new variable,

  • the difference between mpg_h and mpg_c for each car

  • Software (R) does a one sample t-test on those differences.

• Below are the paired t-test and the one sample t-test on the differences

t.test(hwy, cty, alternative="greater", 
       conf.level=.95, paired=T) 

    Paired t-test

data:  hwy and cty
t = 16.913, df = 30, p-value < 0.00000000000000022
alternative hypothesis: true mean difference is greater than 0
95 percent confidence interval:
 6.181461      Inf
sample estimates:
mean difference 
       6.870968 

mpg_diff <- euro_cars$mpg_h-euro_cars$mpg_c
t.test(mpg_diff, alternative="greater")

    One Sample t-test

data:  mpg_diff
t = 16.913, df = 30, p-value < 0.00000000000000022
alternative hypothesis: true mean is greater than 0
95 percent confidence interval:
 6.181461      Inf
sample estimates:
mean of x 
 6.870968 

💥 Lecture 18 In-class Exercises - Q6 💥

Note that the paired t-test output DOES NOT give the mean for each group.

Instead, the paired t-test output shows the mean difference.

Question 6: What is the average difference in mileage between city driving and highway driving for all of these vehicles?

Round to two decimal places.

Introduction to HW 6

• For the remainder of class we will look at HW 6.

• I will also make some short videos but using R will not be the primary challenge for this assignment.

• This assignment includes many multiple choice questions that require an understanding of

  • what each hypothesis test is testing.
  • how to correctly interpret the results.

Key Points from Today

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

• For all hypothesis tests, the hypotheses can be specified three ways:

  • two-sided, left-sided, right-sided
  • This specification matches the ALTERNATIVE Hypothesis, \(H_{A}\).

• In addition to one sample tests, there are two two types of two sample t-tests:

  • tests of two independent groups
  • paired t-tests