Chapter 13 Comparing Two Population Parameters

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Chapter 13Comparing Two Population Parameters

• AP Statistics
• Hamilton and Mann

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Lipitor or Pravachol

• Which drug is more effective at lowering bad
  cholesterol?
• To figure this out, researchers designed a study
  they called PROVE-IT.
• They used 4000 people with heart disease as
  subjects. These people were randomly assigned to
  one of two treatment groups Lipitor or
  Pravachol.
• At the end of the study, researchers compared the
  mean bad cholesterol levels for the two groups.
  For Pravachol it was 95 mg/dl versus 62 mg/dl
  for Lipitor. Is this difference statistically
  significant?
• This is a question about comparing two means.

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Lipitor or Pravachol

• The researchers also compared the proportion of
  subjects in each group who died, had a heart
  attack, or suffered other serious consequences
  within two years.
• For Pravachol, the proportion was 0.263 and for
  Lipitor it was 0.224. Is this a statistically
  significant difference?
• This is a question about comparing two
  proportions.

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Success vs. Failure in Business

• How do small businesses that fail differ from
  small businesses that succeed?
• Business school researchers compared the asset
  liability ratios of two samples of firms started
  in 2000, one sample of failed businesses and one
  of firms that are still going after two years.
• This observational study compares two random
  samples, one from each of two different
  populations.

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Two-Sample Problems

• Comparing two populations or two treatments is
  one of the most common situations encountered in
  statistical practice. We call such situations
  two-sample problems.

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Two-Sample Problems

• A two-sample problem can arise from a randomized
  comparative experiment that randomly divides
  subjects into two groups and exposes each group
  to a different treatment, like the PROVE-IT
  Study.
• Comparing random samples separately selected from
  two populations, like the successful and failed
  small businesses, is also a two-sample problem.
• Unlike the matched pairs designs studied earlier,
  there is no matching of units in the two samples
  and two samples can be of different sizes.
• Inference procedures for two-sample data differ
  from those of matched pairs.

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Comparing Means and Proportions

• Who is more likely to binge drink male or female
  college students?
• This is obviously a two-sample problem because we
  are comparing the population of male college
  students to female college students.
• To conduct this study, the Harvard School of
  Public Health surveyed random samples of male and
  female undergraduates at four-year colleges and
  universities about their drinking behaviors.
• This observational study was designed to compare
  the proportion of undergraduate males who binge
  drink with the proportion of undergraduate
  females who binge drink.

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Comparing Means and Proportions

• A bank wants to know which of two incentive plans
  will most increase the use of its credit cards.
• We are comparing the effect of two different
  treatments here, so it is a two-sample problem.
• It offers each incentive to a random sample of
  credit card customers and compares the amount
  charged during the following six months.
• This is a randomized experiment designed to
  compare the mean amount spent under each of the
  two incentive treatments.

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Chapter 13 Section 1

• Comparing Two Means
• HW 13.1, 13.2, 13.4, 13.6, 13.8, 13.10, 13.11,
  13.14, 13.16

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Comparing Two Means

• We can examine two-sample data graphically by
  comparing dotplots or stempots (for small
  samples) and boxplots or histograms (for large
  samples).
• Now we will apply the ideas of formal inference
  in this setting.
• When both population distributions are symmetric,
  and especially when they are approximately
  Normal, a comparison of the mean responses in the
  two populations is the most common goal of
  inference.

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Notation
Parameters Parameters Statistics Statistics
Population Variable Mean Standard Deviation Sample Size Mean Standard Deviation
1 x1 µ1 ?1 n1 s1
2 x2 µ2 ?2 n2 s2

• There are four unknown parameters, the two means
  and the two standard deviations.
• We want to compare the two population means,
  either by giving a confidence interval for their
  difference µ1 - µ2 or by testing the hypothesis
  of no difference, H0µ1 µ2.
• We use the sample means and standard deviations
  to estimate the unknown parameters.

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Calcium and Blood Pressure

• Does increasing the amount of calcium in our diet
  reduce blood pressure?
• An examination of a large number of people
  revealed a relationship between calcium intake
  and blood pressure. The relationship was
  strongest for black men. As a result,
  researchers designed a randomized comparative
  experiment.
• The subjects were 21 healthy black men. A
  randomly chosen group of 10 of the men received
  calcium supplements for 12 weeks. The other 11
  men received a placebo pill that looked similar
  for the 12 weeks.

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Calcium and Blood Pressure

• The response variable is the decrease in systolic
  blood pressure for a subject after 12 weeks. An
  increase appears as a negative response.
• Group 1 will be the calcium group and Group 2
  will be the placebo group. Here are the data.
• Here are the summary statistics.

Group 1 Calcium Group Group 1 Calcium Group Group 1 Calcium Group Group 1 Calcium Group Group 1 Calcium Group Group 1 Calcium Group Group 1 Calcium Group Group 1 Calcium Group Group 1 Calcium Group Group 1 Calcium Group Group 1 Calcium Group
7 -4 18 17 -3 -5 1 10 11 -2
Group 2 Placebo Group Group 2 Placebo Group Group 2 Placebo Group Group 2 Placebo Group Group 2 Placebo Group Group 2 Placebo Group Group 2 Placebo Group Group 2 Placebo Group Group 2 Placebo Group Group 2 Placebo Group Group 2 Placebo Group
-1 12 -1 -3 3 -5 5 2 -11 -1 -3
Group Treatment n s
1 Calcium 10 5.000 8.743
2 Placebo 11 -0.273 5.901
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Calcium and Blood Pressure

• Notice that the calcium group experienced a drop
  in blood pressure, while the placebo
  group shows a small increase,
  Is this good evidence that calcium decreases
  blood pressure in the entire population of
  healthy black men more than a placebo does?
• This example fits the two-sample setting because
  we have a separate sample from each treatment and
  we have not attempted to match them.
• Since we are testing a claim, we will conduct a
  significance test and follow the Inference
  Toolbox.

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Calcium and Blood Pressure

• Step 1 Hypotheses We write the hypotheses in
  terms of the mean decreases we would see in the
  entire population µ1 of black men taking calcium
  for 12 weeks and µ2 for black men taking the
  placebo for 12 weeks. There are two possible
  hypotheses or

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Calcium and Blood Pressure

• Step 2 Conditions We do not know the name of
  the test, but we know the conditions we must
  check to compare two means.
• SRS The 21 subjects are not an SRS. Therefore,
  we may not be able to generalize our findings to
  all healthy black men. Since we randomly
  assigned treatments, however, any differences can
  be attributed to the treatments themselves.
• Normality Since we have small samples, we must
  look at a boxplot and histogram for both samples.
  There are no serious problems (outliers or
  serious departure from Normality).
• Independence Since we randomized the
  treatments, we can safely assume that the calcium
  and placebo are two independent samples.

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Calcium and Blood Pressure

• The natural estimator of the difference µ1 - µ2
  is the difference between the sample means
• This statistic measures the average advantage of
  calcium over the placebo. In order to use this,
  however, we need to know about its sampling
  distribution. In other words, we need to know
  what the mean and standard deviation would be for
  the population of differences if we took repeated
  samples many times.

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The Two-Sample z Statistic

• Here are the facts about the sampling
  distribution of the difference
  between the two sample means of independent SRSs.
• Therefore,
• If both populations are Normal, then the
  distribution of is also Normal
  with

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Two-Sample z Statistic

• When the statistic has a Normal
  distribution, we can standardize it to obtain a
  standard Normal z statistic.

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Two-Sample z Statistic

• In the very unlikely case that we know both
  population standard deviations, the two-sample z
  statistic is what we would use to conduct
  inference about
• Since we rarely know one, much less two,
  population standard deviations, we are going to
  move immediately to the more useful t procedures.

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Two-Sample t Procedures

• Because we dont know the population standard
  deviations, we estimate them with the standard
  deviations from our two samples.
• The result is the standard error, or estimated
  standard deviation, of the difference in sample
  means
• We then standardize our estimate
  the result if the two-sample t statistic

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Two-Sample t Procedures

• The statistic t has the same interpretation as
  any z or t statistic it says how far
  is from its mean in standard deviation units.
• The two-sample t statistic has approximately a t
  distribution. It does not have exactly a t
  distribution even if the populations are both
  exactly Normal. The approximation is very close
  though.
• There is a catch we must use a messy formula to
  calculate the degrees of freedom. Often, the
  degrees of freedom are not whole numbers.

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Two-Sample t Procedures

• There are two practical options for using the
  two-sample t procedures
• With technology, use the statistic t with
  accurate critical values from the approximating t
  distribution.
• Without technology, use the statistic t with
  critical values from the t distribution with
  degrees of freedom equal to the smaller of n1 1
  and n2 1. These procedures are always
  conservative for any two Normal populations.
• Technology will obviously use method 1.
• We are going to start by looking at how to do
  method 2.

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Two-Sample t Procedures

• These two-sample t procedures always err on the
  safe side, reporting higher P-values and lower
  confidence than may actually be true. The gap
  between what is reported and the truth is
  actually quite small unless the sample sizes are
  both small and unequal.
• As the sample sizes increase, probability values
  based on t with degrees of freedom equal to the
  smaller of n1 1 and n2 1 become more
  accurate.
• Lets complete our calcium and blood pressure
  problem from earlier.

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Calcium and Blood Pressure

• Here are the summary statistics again.
• Step 3 Calculations
• Since it was a one-sided test, we are looking for
  the probability being 1.604 or greater when we
  have 9 degrees of freedom. From the table, it is
  between 0.05 and 0.10.

Group Treatment n s
1 Calcium 10 5.000 8.743
2 Placebo 11 -0.273 5.901
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Calcium and Blood Pressure

• Step 4 Interpretation
• The experiment provides some evidence that
  calcium reduces blood pressure, but the evidence
  falls short of the traditional 5 and 1 levels
  of significance. We would fail to reject H0 at
  both significance levels.

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Creating a Confidence Interval

• We can estimate the difference in mean decreases
  in blood pressure for the hypothetical calcium
  and placebo populations using a two-sample t
  interval.
• We have already checked all of the conditions.
• Recall
• Since the 90 confidence interval includes 0, we
  cannot reject H0µ1 µ2 0 against the
  two-sided alternative at the a 0.10 level of
  significance.

Group Treatment n S
1 Calcium 10 5.000 8.743
2 Placebo 11 -0.273 5.901
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Sample Size Matters

• Sample sizes strongly influence the P-value of a
  test.
• A result that fails to be significant at a
  specified level a in a small sample may be
  significant in a larger sample.
• For instance, the difference of 5.273 in the mean
  systolic blood pressures between our two groups
  was not significant. In a larger study with more
  subjects, they were able to obtain a P-value of
  0.008.

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Robustness Again

• The two-sample t procedures are more robust than
  the one-sample t procedures, particularly when
  the distributions are not symmetric.
• When the sizes of the two samples are equal and
  the two populations being compared have
  distributions with similar shapes, probability
  values from the t table are quite accurate for a
  broad range of distributions for samples as small
  as 5. When the populations have different
  shapes, larger samples are needed.

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Robustness Again

• As a guide to practice, adapt the guidelines on
  p. 655 for the use of one-sample t procedures to
  two-sample t procedures by replacing sample
  size with the sum of the sample sizes as long
  as both samples are at least 5.
• These guidelines err on the side of safety,
  especially when the two-samples are of equal
  size.
• Whenever possible, try to make both samples the
  same size. Two-sample procedures are most robust
  against non-Normality when the sample sizes are
  equal and the conservative P-values are most
  accurate.

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Software Approximations for the DF

• The t procedures remain exactly as before except
  that we use the t distribution with df given by
  the formula in the box above to give critical
  values and find P-values.

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Calcium and Blood Pressure

• Here are the summary statistics again.
• For improved accuracy, lets calculate the df
  given by the formula on the prior slide.

Group Treatment n s
1 Calcium 10 5.000 8.743
2 Placebo 11 -0.273 5.901
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• Notice that the P-value here is 0.064 compared to
  the 0.0716 we got from the conservative approach.

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Degrees of Freedom

• The formula from the box will always give us df
  at least as large as the smaller of the two
  samples and never bigger than n1 n2 -2.
• The number of degrees of freedom is generally not
  a whole number. Since the table only has whole
  numbers, we will need to use technology to do
  these calculations easily.
• Lets do the Calcium and Blood Pressure problem
  on the calculator!
• We should use the calculator to do these
  calculations from now on!

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DDT Poisoning

• Poisoning by the pesticide DDT causes convulsions
  in humans and other mammals. Researchers seek to
  understand how the convulsions are caused. In a
  randomized comparative experiment, the compared 6
  white rats poisoned with DDT with a control group
  of 6 unpoisoned rats. Electrical measurements of
  nerve activity are the main clue to the nature of
  DDT poisoning. When a nerve is stimulated, its
  electrical response shows a sharp spike followed
  by a much smaller second spike. The experiment
  found that the second spike is larger in rats fed
  DDT than in normal rats.

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DDT Poisoning

• The researchers measured the height (or
  amplitude) of the second spike as a percent of
  the first spike when a nerve in the rats leg was
  stimulated.
• For the poisoned rats the results were
• For the control group the results were
• Lets conduct a significance test at the 0.05
  significance level to determine if there is a
  difference using the calculator.

12.207 16.869 25.050 22.429 8.456 20.589
11.074 9.686 12.064 9.351 8.182 6.642
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DDT Poisoning

• Step 1 Hypotheses
• We want to compare the mean height µ1 of the
  second-spike electrical response in rats fed DDT
  with the mean height µ2 of the second-spike
  electrical response in the population of normal
  rats. Or

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DDT Poisoning

• Step 2 Conditions Since both population
  standard deviations are unknown we need to
  conduct a 2-sample t test.
• SRS By randomly assigning the rats to the
  treatments, we can conclude that differences are
  a result of the treatment. The researchers are
  willing to assume that the two samples of rats
  represent an SRS.
• Normality We dont know if the populations are
  Normal and do not have a large enough sample. We
  must look at a boxplot and histogram. No
  outliers or heavy skewness.
• Independence Due to the random assignment, the
  researchers can treat the two groups as
  independent.

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DDT Poisoning

• Step 3 Calculations
• Since it is a two-sided hypothesis, we must find
  the probability that we are less than -2.99 or
  greater than 2.99.
• The degrees of freedom are df 5.9 and the
  P-value from t(5.9) distribution is 0.0246.
• Step 4 Conclusion
• Since 0.0246 is less than the significance level
  of 0.05, we reject the null hypothesis and
  conclude that there is sufficient evidence to
  conclude that the height of the second-spike
  electrical response in rats fed DDT differs from
  that of normal rats.

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Pooled Two-Sample t Procedures

• Do not use them.
• If a printout says pooled, do not use that.
  Instead use the one that says unpooled.
• On the calculator, always do No for pooled.
• If you want more information you can read it on
  p. 800.

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Chapter 13 Section 2

• Comparing Two Proportions
• HW 13.26, 13.27, 13.28, 13.29, 13.30, 13.32,
  13.33, 13.38

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Prayer and In Vitro Pregnancy

• Some women want to have children but cannot for
  medical reasons. One option for these women is
  in vitro fertilization. About 28 of women who
  undergo in vitro fertilization get pregnant. Can
  praying for these women help increase the
  pregnancy rate?
• Researchers developed an experiment to help
  answer this question. (Why not just survey women
  who have already gone through in vitro to find
  out if a higher percentage of women who were
  prayed for got pregnant?)

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Prayer and In Vitro Pregnancy

• A large group of women who were about to undergo
  in vitro fertilization served as the subjects.
  Each subject was randomly assigned to the
  treatment group (prayed for by people who did not
  know them) or a control group (no prayer).
• The results 44 of the 88 women (50) got
  pregnant in the treatment (prayer) group while
  only 21 out of 81 got pregnant in the control
  group.
• This seems like a large difference, but is it
  statistically significant?

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Two-Sample Proportions

• We will use notation that is similar to what we
  used for two-sample means. We still want to
  compare two groups, Population 1 and Population
  2.
• Here is the notation
• We compare the populations by doing inference
  about the difference p1 - p2 between the
  population proportions.
• The statistic that estimates this difference is

Population Population Proportion Sample Size Sample Proportion
1 p1 n1
2 p2 n2
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Does Preschool Help?

• To study the long-term effects of preschool
  programs for poor children, the High/Scope
  Educational Research Foundation has followed two
  groups of Michigan children since early
  childhood.
• Group 1 Control Group 61 children from
  population 1, poor children with no preschool
• Group 2 Treatment Group 62 children from
  population 2, poor children with preschool as 3-
  and 4-year-olds.
• Both groups were from the same area and had
  similar backgrounds.
• So our sample sizes are n1 61 and n2 62.

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Does Preschool Help?

• One response variable of interest is the need for
  social services as adults. In the past ten
  years, 49 of the control sample and 38 of the
  preschool sample had needed social services. So
  the sample proportions are
• To see if the study provides significant evidence
  that preschool reduces the later need for social
  services, we are going to create a 95 confidence
  interval.

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Does Preschool Help?

• To estimate how large the reduction is, we give a
  confidence interval for the difference.
• Both the test and the confidence interval start
  with the difference in the sample proportions
• This means we need to know the sampling
  distribution of
• So lets look at that now!

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Sampling Distribution of .

• Both are random variables because
  their values would vary if we took repeated
  samples of the same size.
• In Chapter 7, we learned that if X and Y are any
  two random variables then
• In Chapter 9, we learned that

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Sampling Distribution of .

• Using all of this information, we can find the
  mean and standard deviation of
• If the two sample proportions are independent,
• Thus

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Sampling Distribution of .

• As far as the shape, the distribution will be
  approximately normal when both of the
  distributions are approximately Normal.
• In other words,
• Actually, we are safe performing significance
  tests about as long as all of these
  values are greater than 5.
• The distribution of is on the next
  graph.

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Sampling Distribution of .
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Sampling Distribution of .

• The standard deviation of involves
  the unknown parameters p1 and p2.
• Just like in Chapter 12, we must replace these by
  estimates in order to do inference.
• Just like in Chapter 12, we do this a bit
  differently for confidence intervals and
  significance tests.

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Confidence Intervals for .

• To obtain a confidence interval, replace p1 and
  p2 in the expression for with the
  sample proportions.
• The result is the standard error of the statistic
  
• The confidence interval again has the form

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Does Preschool Help?

• Here is a summary of the information from the
  preschool problem we discussed earlier.
• We setup our hypotheses earlier. So we have
  already done Step 1. Here are the Hypotheses as
  a reminder. or

Population Population Description Sample Size Sample Proportion
1 Control n1 61
2 Preschool n2 62
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Does Preschool Help

• Step 2 Conditions We are going to construct a
  two-proportion z interval.
• SRS We were not told how the children were
  selected, so we must be cautious when drawing
  conclusions.
• Normality - Since all are at least 5 we can
  assume Normality.
• Independence We are fairly certain that there
  are at least 610 poor children who did not attend
  preschool and 620 poor children who did attend
  preschool in our populations of interest.

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Does Preschool Help

• Step 3 Calculations
• Step 4 Interpretation
• We are 95 confident that the percent needing
  social services is between 3.3 and 34.7 lower
  among those who attended preschool. The interval
  is wide because of the small sample sizes. Also,
  our results may be questionable due to the fact
  that the samples may not have been SRSs.

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Significance Tests for .

• Observed differences in sample proportions may
  reflect a difference in the populations, or it
  may just be due to variation due to random
  sampling.
• Significance tests help us to determine if the
  difference we see is really there or just chance
  variation.
• The null hypothesis will always say that there is
  no difference in the two populations. Hence
• The alternative hypothesis will always say what
  kind of difference we expect.

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Significance Tests for .

• To conduct a significance test, we must
  standardize to get a z statistic.
• If H0 is true, all the observations in both
  samples come from a single population.
• So, instead of estimating p1 and p2 separately,
  we combine the two samples and use the overall
  sample proportion to estimate the single
  population parameter p.

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Significance Tests for .

• We call this single proportion the combined
  sample proportion. It is
• Now, we use in place of both
  in the expression for the standard error of
• This yields a z statistic that has the standard
  Normal distribution when H0 is true.

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Cholesterol and Heart Attacks

• High levels of cholesterol in the blood are
  associated with higher risk of heart attacks.
  Does using a drug to lower blood cholesterol
  reduce heart attacks?
• The Helsinki Heart Study looked at this question
  by randomly assigning middle-aged men to one of
  two treatments 2051 men took the drug
  gemfibrozil to reduce their cholesterol levels,
  and a control group of 2030 men took a placebo.
• During the next 5 years, 56 men in the
  gemfibrozil group and 84 men in the control group
  had heart attacks.

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Cholesterol and Heart Attacks

• Is the apparent benefit of gemfibrozil
  statistically significant?
• To answer this question, we need to conduct a
  significance test.
• To conduct a significance test we need So
  lets find

Population Population Description Sample Size Sample Proportion
1 Gemfibrozil n1 2051
2 Control n2 2030
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Cholesterol and Heart Attacks

• Step 1 Hypotheses We want to use this
  comparative randomized experiment to draw
  conclusions about p1, the proportion of
  middle-aged men who would suffer heart attacks
  after taking gemfibrozil, and p2, the proportion
  of middle-aged men who would suffer heart attacks
  if they only took a placebo. We hope to show
  that gemfibrozil reduces heart attacks, so we
  have a one-sided alternative.

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Cholesterol and Heart Attacks

• Step 2 Conditions - We are going to conduct a
  two-proportion z test.
• SRS Since the data come from a comparative
  randomized experiment, we meet this condition.
  This will allow us to conclude that the treatment
  caused the differences we observe. Since the men
  in the experiment were not randomly selected, we
  may not be able to generalize our results to the
  population of all middle-aged men.
• Normality We must use to check for
  Normality since we are assuming that both
  proportions are the same. So
• Independence Due to the random assignment of
  men, the two groups of men can be viewed as
  independent samples.

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Cholesterol and Heart Attacks

• Step 3 Calculations
• We believed it would decrease heart attacks, so
  we need the probability that we are less than or
  equal to -2.47.

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Cholesterol and Heart Attacks

• Step 4 Interpretation Since our P-value
  (0.0068) is less than 0.01, our results are
  significant at the a 0.01 significance level.
  So there is strong evidence that gemfibrozil
  reduced the rate of heart attacks.

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Dont Drink the Water

• The movie A Civil Action tells the story of a
  legal battle that took place in the small town of
  Woburn, Massachusetts. A town well that supplied
  water to East Woburn residents was contaminated
  by industrial chemicals. During the period that
  residents drank the water from this well, a
  sample of 414 births showed 16 birth defects. On
  the west side of Woburn, a sample of 228 babies
  born during the same time period revealed 3 with
  birth defects. The plaintiffs suing the
  companies responsible for the contamination
  claimed that these data show that the rate of
  birth defects was significantly higher in East
  Woburn, where the contaminated well water was in
  use. How strong is the evidence supporting the
  claim? What decision should the judge make?

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Dont Drink the Water

• To conduct a significance test we need So
  lets find
• Step 1 Hypotheses We are interested in seeing
  if there is a difference in the proportion of
  birth defects between East and West Coburn.

Population Population Description Sample Size Sample Proportion
1 East Coburn n1 414
2 West Coburn n2 228
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Dont Drink the Water

• Conditions We are going to conduct a
  Two-Proportion z test.
• SRS We dont know that they are SRSs, but we
  will treat them as SRSs.
• Normality We must check our rules.
  Since each is larger than 5, it
  is approximately Normal.
• Independence We must assume that both
  populations are at least 10 times as large as the
  sample of babies.

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Dont Drink the Water

• Step 3 - Calculations
• The P-value would be the probability that we
  would be 1.82 or greater.
• Step 4 Interpretation
• Since the P-value (0.0344) is smaller than the
  usual level of significance of 0.05, we reject
  the null hypothesis and conclude that there is
  reason to believe that the proportion of birth
  defects was higher in East Coburn.

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