1. Comparing Means from Two Samples
and
One-Sample Inference for Proportions
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2. Stat 111 - Lecture 14 - Two Means
Administrative Notes
Homework 5 is posted on website
Due Wednesday, July 1st
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3. Stat 111 - Lecture 14 - Two Means
Outline
Two Sample Z-test (known variance)
Two Sample t-test (unknown variance)
Matched Pair Test and Examples
Tests and Intervals for Proportions (Chapter 8)
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4. Comparing Two Samples
Up to now, we have looked at inference for one sample of continuous data
Our next focus in this course is comparing the data from two different samples
For now, we will assume that these two different samples are independent of each other and come from two distinct populations
Population 1:1 , 1
Population 2: 2 , 2
Sample 1: , s1
Sample 2: , s2
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5. Blackout Baby Boom Revisited
Nine months (Monday, August 8th) after Nov blackout, NY Times claimed an increased birth rate
Already looked at single two-week sample: found no significant difference from usual rate (430 births/day)
What if we instead look at difference between weekends and weekdays?
Sun
Mon
Tue
Wed
Thu
Fri
Sat
452
470
431
448
467
377
344
449
440
457
471
463
405
453
499
461
442
444
415
356
519
443
418
394
399
451
468
432
Weekdays
Weekends
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6. Two-Sample Z test
We want to test the null hypothesis that the two populations have different means
H0: 1 = 2 or equivalently, 1 - 2 = 0
Two-sided alternative hypothesis: 1 - 2  0
If we assume our population SDs 1 and 2 are known, we can calculate a two-sample Z statistic:
We can then calculate a p-value from this Z statistic using the standard normal distribution
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7. Two-Sample Z test for Blackout Data
To use Z test, we need to assume that our pop. SDs are known: 1 = s1 = 21.7 and 2 = s2 = 24.5
From normal table, P(Z > 7.5) is less than , so our p-value = 2  P(Z > 7.5) is less than
Conclusion here is a significant difference between birth rates on weekends and weekdays
We don’t usually know the population SDs, so we need a method for unknown 1 and 2
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8. Stat 111 - Lecture 14 - Two Means
Two-Sample t test
We still want to test the null hypothesis that the two populations have equal means (H0: 1 - 2 = 0)
If 1 and 2 are unknown, then we need to use the sample SDs s1 and s2 instead, which gives us the two-sample T statistic:
The p-value is calculated using the t distribution, but what degrees of freedom do we use?
df can be complicated and often is calculated by software
Simpler and more conservative: set degrees of freedom equal to the smaller of (n1-1) or (n2-1)
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9. Two-Sample t test for Blackout Data
To use t test, we need to use our sample standard deviations s1 = 21.7 and s2 = 24.5
We need to look up the tail probabilities using the t distribution
Degrees of freedom is the smaller of n1-1 = 22
or n2-1 = 7
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10. Stat 111 - Lecture 14 - Two Means
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11. Two-Sample t test for Blackout Data
From t-table with df = 7, we see that
P(T > 7.5) <
If our alternative hypothesis is two-sided, then we know that our p-value < 2  = 0.001
We reject the null hypothesis at -level of 0.05 and conclude there is a significant difference between birth rates on weekends and weekdays
Same result as Z-test, but we are a little more conservative
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12. Two-Sample Confidence Intervals
In addition to two sample t-tests, we can also use the t distribution to construct confidence intervals for the mean difference
When 1 and 2 are unknown, we can form the following 100·C% confidence interval for the mean difference 1 - 2 :
The critical value tk* is calculated from a t distribution with degrees of freedom k
k is equal to the smaller of (n1-1) and (n2-1)
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13. Confidence Interval for Blackout Data
We can calculate a 95% confidence interval for the mean difference between birth rates on weekdays and weekends:
We get our critical value tk* = is calculated from a t distribution with 7 degrees of freedom, so our 95% confidence interval is:
Since zero is not contained in this interval, we know the difference is statistically significant!
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14. Stat 111 - Lecture 14 - Two Means
Matched Pairs
Sometimes the two samples that are being compared are matched pairs (not independent)
Example: Sentences for crack versus powder cocaine
We could test for the mean difference between X1 = crack sentences and X2 = powder sentences
However, we realize that these data are paired: each row of sentences have a matching quantity of cocaine
Our t-test for two independent samples ignores this relationship
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15. Stat 111 - Lecture 14 - Two Means
Matched Pairs Test
First, calculate the difference d = X1 - X2 for each pair
Then, calculate the mean and SD of the differences d
Quantity
Sentences
Crack X1
Powder X2
Difference
d = X1 - X2 5
70.5 12
58.5 25
87.5 18
69.5
100
136 30
106.0
200
169.5 37
132.5
500
211.5
141.0
2000
264
176.5
5000
128.0
50000
52.5
150000
0.0
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16. Stat 111 - Lecture 14 - Two Means
Matched Pairs Test
Instead of a two-sample test for the difference between X1 and X2, we do a one-sample test on the difference d
Null hypothesis: mean difference between the two samples is equal to zero
H0 : d= versus Ha : d 0
Usual test statistic when population SD is unknown:
p-value calculated from t-distribution with df = 8
P(T > 5.24) < so p-value < 0.001
Difference between crack and powder sentences is statistically significant at -level of 0.05
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17. Matched Pairs Confidence Interval
We can also construct a confidence interval for the mean differenced of matched pairs
We can just use the confidence intervals we learned for the one-sample, unknown  case
Example: 95% confidence interval for mean difference between crack and powder sentences:
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18. Summary of Two-Sample Tests
Two independent samples with known 1 and 2
We use two-sample Z-test with p-values calculated using the standard normal distribution
Two independent samples with unknown 1 and 2
We use two-sample t-test with p-values calculated using the t distribution with degrees of freedom equal to the smaller of n1-1 and n2-1
Also can make confidence intervals using t distribution
Two samples that are matched pairs
We first calculate the differences for each pair, and then use our usual one-sample t-test on these differences
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19. One-Sample Inference for Proportions
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20. Stat 111 - Lecture 14- One-Sample Proportions
Revisiting Count Data
Chapter 6 and 7 covered inference for the population mean of continuous data
We now return to count data:
Example: Opinion Polls
Xi = 1 if you support Obama, Xi = 0 if not
We call p the population proportion for Xi = 1
What is the proportion of people who support the war?
What is the proportion of Red Sox fans at Penn?
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21. Inference for population proportion p
We will use sample proportion as our best estimate of the unknown population proportion p
where Y = sample count
Tool 1: use our sample statistic as the center of an entire confidence interval of likely values for our population parameter
Confidence Interval : Estimate ± Margin of Error
Tool 2: Use the data to for a specific hypothesis test
Formulate your null and alternative hypotheses
Calculate the test statistic
Find the p-value for the test statistic
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22. Distribution of Sample Proportion
In Chapter 5, we learned that the sample proportion technically has a binomial distribution
However, we also learned that if the sample size is large, the sample proportion approximately follows a Normal distribution with mean and standard deviation:
We will essentially use this approximation throughout chapter 8, so we can make probability calculations using the standard normal table
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23. Confidence Interval for a Proportion
We could use our sample proportion as the center of a confidence interval of likely values for the population parameter p:
The width of the interval is a multiple of the standard deviation of the sample proportion
The multiple Z* is calculated from a normal distribution and depends on the confidence level
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24. Confidence Interval for a Proportion
One Problem: this margin of error involves the population proportion p, which we don’t actually know!
Solution: substitute in the sample proportion for the population proportion p, which gives us the interval:
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25. Example: Red Sox fans at Penn
What proportion of Penn students are Red Sox fans?
Use Stat 111 class survey as sample
Y = 25 out of n = 192 students are Red Sox fans so
95% confidence interval for the population proportion:
Proportion of Red Sox fans at Penn is probably between 8% and 18%
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26. Hypothesis Test for a Proportion
Suppose that we are now interested in using our count data to test a hypothesized population proportion p0
Example: an older study says that the proportion of Red Sox fans at Penn is 0.10.
Does our sample show a significantly different proportion?
First Step: Null and alternative hypotheses
H0: p = vs Ha: p 0.10
Second Step: Test Statistic
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27. Hypothesis Test for a Proportion
Problem: test statistic involves population proportion p
For confidence intervals, we plugged in sample proportion but for test statistics, we plug in the hypothesized proportion p0 :
Example: test statistic for Red Sox example
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28. Hypothesis Test for a Proportion
Third step: need to calculate a p-value for our test statistic using the standard normal distribution
Red Sox Example: Test statistic Z = 1.39
What is the probability of getting a test statistic as extreme or more extreme than Z = 1.39? ie. P(Z > 1.39) = ?
Two-sided alternative, so p-value = 2P(Z>1.39) = 0.16
We don’t reject H0 at a =0.05 level, and conclude that Red Sox proportion is not significantly different from p0=0.10
prob = 0.082
Z = 1.39
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29. Stat 111 - Lecture 14- One-Sample Proportions
Another Example
Mass ESP experiment in 1977 Sunday Mirror (UK)
Psychic hired to send readers a mental message about a particular color (out of 5 choices). Readers then mailed back the color that they “received” from psychic
Newspaper declared the experiment a success because, out of responses, they received 521 correct ones ( )
Is the proportion of correct answers statistically different than we would expect by chance (p0 = 0.2) ?
H0: p= vs. Ha: p 0.2
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30. Stat 111 - Lecture 14- One-Sample Proportions
Mass ESP Example
Calculate a p-value using the standard normal distribution
Two-sided alternative, so p-value = 2P(Z>2.43) = 0.015
We reject H0 at a =0.05 level, and conclude that the survey proportion is significantly different from p0=0.20
We could also calculate a 95% confidence interval for p:
prob =
Z = 2.43
Interval doesn’t contain 0.20
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31. Stat 111 - Lecture 14- One-Sample Proportions
Margin of Error
Confidence intervals for proportion p is centered at the sample proportion and has a margin of error:
Before the study begins, we can calculate the sample size needed for a desired margin of error
Problem: don’t know sample prop. before study begins!
Solution: use which gives us the maximum m
So, if we want a margin of error less than m, we need
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32. Margin of Error Examples
Red Sox Example: how many students should I poll in order to have a margin of error less than 5% in a 95% confidence interval?
We would need a sample size of 385 students
ESP example: how many responses must newspaper receive to have a margin of error less than 1% in a 95% confidence interval?
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33. Stat 111 - Lecture 14- One-Sample Proportions
Next Class - Lecture 15
Two-Sample Inference for Proportions
Moore, McCabe and Craig: Section 8.2
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