TRUE OR FALSE: When conducting a two-tailed two sample hypothesis test of means, we can only tell if two population means are significantly different, not which one is larger (or smaller).
Two-sided Two Sample Hypothesis Test
\(H_{0}: \mu_{1} = \mu_{2}\)
\(H_{A}: \mu_{1} \neq \mu_{2}\)
Review of One Sample Hypothesis Tests of Proportions
Would you like to see the changing of the clocks eliminated, so people no longer change their clocks twice per year?
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.
If we test these data, what are the null and alternative hpotheses:
\(H_{0}: P_{YES} = P_{NO}\) - 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.
Specify alpha (\(\alpha\)) as 0.05 unless we have a specific reason to choose a different alpha.
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
\(H_{0}: P_{YES} = 0.5\)
\(H_{A}: P_{YES} \neq 0.5\)
How do we interpret the outcome of this hypothesis test?
Contingency Tables to Examine Proportions in Multiple Categories
Question: Are these disparities in opinions about daylight savings consistent among age groups?
We can examine this question using tables, plots, and hypothesis tests.
A Contingency Table is 2 x 2 or larger and allows us to subdivide count data by categories
Commonly used in market research to understand opinions by category
Example: How do Gen Z (18-29) and Millennial adults feel (30-44) about daylight savings?
Should the USA Eliminate Daylight Savings Clock Changes
Yes
No/Not Sure
Ages 18-29
99
98
Ages 30-44
129
105
Comparing Contingency Tables and Plots
Contingency tables and bar plots are two effective ways to examine these data
Should the USA Eliminate Daylight Savings Clock Changes
Yes
No/Not Sure
Ages 18-29
99
98
Ages 30-44
129
105
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.
Questions we will answer:
What is the p-value from this test?
Do we Reject or Fail to Reject the Null Hypothesis?
What do we conclude about the opinions of these two age groups?
💥 Lecture 19 In-class Exercises - Q3-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-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?
Should the USA Eliminate Daylight Savings Clock Changes
Yes
No/Not Sure
Ages 18-44
228
205
Ages 45-64
201
118
Column and Row Percentages
Original Data
Should the USA Eliminate Daylight Savings Clock Changes
Yes
No/Not Sure
Ages 18-44
228
205
Ages 45-64
201
118
Row Percentages: Percentages of each age group that said ‘Yes’ or ‘No’.
Row Percentages
Yes
No/Not Sure
Ages 18-44
52.66
47.34
Ages 45-64
63.01
36.99
Column percentages: Percentages of Yes/No opinions in each age group.
Column Percentages
Yes
No/Not Sure
Ages 18-44
53.15
63.47
Ages 45-64
46.85
36.53
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 proportions?
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.
To submit an Engagement Question or Comment about material from Lecture 19: Submit it by midnight today (day of lecture).
