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).
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
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 (Usually a two-sided tests):
\(H_{0}: P_{18-29} = P_{30-44}\)
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?
What do we conclude about the opinions of these two age groups?
Question 6:
(Not on Poll Everywhere): 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
