Comparing Means from Two Samples

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Comparing Meansfrom Two Samples
Statistics 111 Lecture 14
and
One-Sample Inference for Proportions
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Administrative Notes

• Homework 5 is posted on website
• Due Wednesday, July 1st

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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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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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Blackout Baby Boom Revisited

• Nine months (Monday, August 8th) after Nov 1965
  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?

Weekdays
Weekends
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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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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 gt 7.5) is less than
  0.0002, so our p-value 2 ? P(Z gt 7.5) is less
  than 0.0004
• Conclusion here is a significant difference
  between birth rates on weekends and weekdays
• We dont usually know the population SDs, so we
  need a method for unknown ?1 and ?2

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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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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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Two-Sample t test for Blackout Data

• From t-table with df 7, we see that
• P(T gt 7.5) lt 0.0005
• If our alternative hypothesis is two-sided, then
  we know that our p-value lt 2 ? 0.0005 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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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 100C 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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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 2.365 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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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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Matched Pairs Test

• First, calculate the difference d X1 - X2 for
  each pair
• Then, calculate the mean and SD of the
  differences d

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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 0 versus Ha ?d? 0
• Usual test statistic when population SD is
  unknown
• p-value calculated from t-distribution with df
  8
• P(T gt 5.24) lt 0.0005 so p-value lt 0.001
• Difference between crack and powder sentences is
  statistically significant at ?-level of 0.05

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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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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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One-Sample Inference for Proportions
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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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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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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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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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Confidence Interval for a Proportion

• One Problem this margin of error involves the
  population proportion p, which we dont actually
  know!
• Solution substitute in the sample proportion
  for the population proportion p, which gives us
  the interval

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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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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 0.10 vs. Ha p? 0.10
• Second Step Test Statistic

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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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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 gt 1.39) ?
• Two-sided alternative, so p-value 2?P(Zgt1.39)
  0.16
• We dont reject H0 at a ?0.05 level, and
  conclude that Red Sox proportion is not
  significantly different from p00.10

prob 0.082
Z 1.39
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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 2355 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 0.2 vs. Ha p? 0.2

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Mass ESP Example

• Calculate a p-value using the standard normal
  distribution
• Two-sided alternative, so p-value 2?P(Zgt2.43)
  0.015
• We reject H0 at a ?0.05 level, and conclude that
  the survey proportion is significantly different
  from p00.20
• We could also calculate a 95 confidence interval
  for p

prob 0.0075
Z 2.43
Interval doesnt contain 0.20
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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 dont 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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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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Next Class - Lecture 15

• Two-Sample Inference for Proportions
• Moore, McCabe and Craig Section 8.2