Review

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Review

• Central Limit Theorem
• and CI for P

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• s sample distribution is narrower than the
  population distribution, by a factor of vn.

Sample means,n subjects
Population, xindividual subjects
m
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Review 95 Confidence Interval for a Proportion
We are 95 confident that p (the true pop
proportion) lies in the interval.
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The best reason ever to study math
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Chapter 22 23

• What Is a Test of Significance?
• Use and Abuse of Statistical Inference

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What is a hypothesis?
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What is a hypothesis?

• Testable
• Falsifiable

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Scientific method
1.Define the question 2.Gather information and
resources 3.Form hypothesis 4.Plan
experiment 5.Do experiment and collect
data 6.Analyze data 7.Interpret data and draw
conclusions that serve as a starting point for
new hypotheses 8.Communicate results
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Using Data to Make Decisions

• Examining Confidence Intervals.
• Hypothesis Tests
• Is the sample data statistically significant, or
  could it have happened by chance?

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Steps for Testing Hypotheses

• Determine the null hypothesis and the
  alternative hypothesis.
• Collect data and summarize with a single number
  called a test statistic.
• Determine how unlikely test statistic would be if
  null hypothesis were true.
• Make a decision.

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Determine the hypotheses.

• Null hypothesishypothesis that says nothing is
  happening, status quo, no relationship, chance
  only.
• Alternative (research) hypothesis hypothesis is
  reason data being collected researcher suspects
  status quo belief is incorrect or that there is a
  relationship between two variables that has not
  been established before.

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Collect data and summarize with a test statistic.

• Decision in hypothesis test based on single
  summary of data the test statistic.
• standard Z score.
• Also will see t, F, r, .

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Determine how unlikely test statistic would be if
null hypothesis true
p-value - If null hypothesis true, how likely to
observe sample results of this magnitude or
larger (in direction of the alternative) just by
chance? P-value often misinterpreted in the
news.
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Make a Decision.

• p-value not small enough to convincingly rule out
  chance.
• We cannot reject the null hypothesis as an
  explanation for the results.
• There is no statistically significant difference
  or relationship evidenced by the data.
• p-value small enough to convincingly rule out
  chance.
• We reject the null hypothesis and accept the
  alternative hypothesis.
• There is a statistically significant difference
  or relationship evidenced by the data.

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Hypothesis test for p
Suppose 60 (0.60) of the population are in favor
of new tax legislation. A pollster takes a
sample of 265 people which results in 175, or
0.66, who are in favor. How likely is it that
we would see a sample proportion as large as 0.66
given the true mean proportion of 0.60? Is this
highly unusual?
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Hypothesis test for p
The true population proportion, p 0.60 Our
sample data yields p-hat 175/ 265 0.66,
From the Rule for Sample Proportions, we know
the potential sample proportions in this
situation follow an approximately normal
distribution, with a mean of 0.60 and a standard
deviation of 0.03.
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Hypothesis test for p
If the sampling distn of p-hat is normal with a
mean of 0.60 and a SD of 0.03, how many SDs
above the mean is 0.66? Recall from the chapter
on the normal distribution, what is the z-score
or standardized score of 0.66?
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Test for p Bell-Shaped Curve of Sample
Proportions (n265)
mean 0.60 S.D. 0.03
2. 5
0.60
0.63
0.57
0.66
0.54
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Test for p continued
Suppose that in the previous question we do not
know for sure that the proportion of the
population who favor the new tax legislation is
60. Instead, this is just the claim of a
politician. From the data collected, we have
discovered that if the claim is true, then the
sample proportion observed falls at about the
98th percentile of possible sample proportions
for that sample size. Should we believe the
claim and conclude that we just observed strange
data, or should we reject the claim? What if the
result fell at the 85th percentile? At the
99.99th percentile?
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Bell-Shaped Curve of Sample Proportions (n265)
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The Null Hypothesis H0

• population parameter equals some value
• status quo
• no relationship
• no change
• no difference in two groups
• etc.
• When performing a hypothesis test, we assume that
  the null hypothesis is true until we have
  sufficient evidence against it

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The Alternative Hypothesis Ha

• population parameter differs from some value
• not status quo
• relationship exists
• a change occurred
• two groups are different
• etc.

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The Hypotheses for Proportions

• Null H0 p p0
• One-sided alternatives
• Ha p gt p0
• Ha p lt p0
• Two-sided alternative
• Ha p ¹ p0

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Example from class dataOur hypotheses do we
have a majority?

• Null H0 p .5
• Alt Ha p gt .5

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Sampling Distribution for Proportions
If numerous simple random samples of size n are
taken, the sample proportions from the
various samples will have an approximately normal
distribution with mean equal to p (the population
proportion) and standard deviation equal to
Since we assume the null hypothesis is true, we
replace p with p0 to complete the test.
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Test Statistic for Proportions
To determine if the observed proportion is
unlikely to have occurred under the assumption
that H0 is true, we must first convert the
observed value to a standardized score
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Test Statistic

• Based on the sample
• n___ (large, so proportions follow normal
  distribution)
• Sample
• standard error of
• (where .50 is p0 from the null hypothesis)
• standardized score (test statistic)
• z (___- 0.50) /SE ____

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P-value

• The P-value is the probability of observing data
  this extreme or more so in a sample of this size,
  assuming that the null hypothesis is true.
• A small P-value indicates that the observed data
  (or relationship) is unlikely to have occurred if
  the null hypothesis were actually true
• The P-value tends to be small when there is
  evidence in the data against the null hypothesis
• The P-value is NOT the probability that the null
  hypothesis is true

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P-value for Testing Proportions

• Ha p gt p0
• When the alternative hypothesis includes a
  greater than gt symbol, the P-value is the
  probability of getting a value as large or larger
  than the observed test statistic (z) value.
• The area in the right tail of the bell curve

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P-value for Testing Proportions

• Ha p lt p0
• When the alternative hypothesis includes a less
  than lt symbol, the P-value is the probability
  of getting a value as small or smaller than the
  observed test statistic (z) value.
• The area in the left tail of the bell curve (the
  same as the percentile value)

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P-value for Testing Proportions

• Ha p ? p0
• When the alternative hypothesis includes a not
  equal to ? symbol, the P-value is twice as
  large as a one-sided test (the sign of z could go
  either way, we just believe there is a
  difference).
• The area in both tails of the bell curve
• double the area in one tail (symmetry)

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P-value
Ha p gt .50
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Decision

• If we think the P-value is too low to believe the
  observed test statistic is obtained by chance
  only, then we would reject chance (reject the
  null hypothesis) and conclude that a
  statistically significant relationship exists
  (accept the alternative hypothesis).
• Otherwise, we fail to reject chance anddo not
  reject the null hypothesis of no relationship
  (result not statistically significant).

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Typical Cut-off for the P-value

• Commonly, P-values less than 0.05 are considered
  to be small enough to reject chance (reject the
  null hypothesis).
• Some researchers use 0.10 or 0.01 as the cut-off
  instead of 0.05.
• This cut-off value is typically referred to as
  the significance level (alpha) of the test.

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Decision

• We do not find the result to be statistically
  significant.
• We fail to reject the null hypothesis. It is
  plausible that there was not a majority.
• or
• We do find the result to be statistically
  significant.
• We reject the null hypothesis. The data supports
  the hypothesized majority.

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• What numerical value gives you the answer to the
  question of how unlikely the test statistic would
  be if the null hypothesis were true?
• The p-value
• The confidence interval
• The sample standard deviation

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Types of errors in decision making
In the courtroom, juries must make a decision
about the guilt or innocence of a defendant.
Which mistake is more serious A. if the jury
claims the suspect is guilty when in fact he or
she is innocent. B. if the jury claims the
suspect is not guilty when in fact he or she is
guilty
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Example A Jury Trial
If on a jury, must presume defendant is innocent
unless enough evidence to conclude is guilty.
Null hypothesis Defendant is innocent. Alternati
ve hypothesis Defendant is guilty.

• Trial held because prosecution believes status
  quo of innocence is incorrect.
• Prosecution collects evidence, like researchers
  collect data, in hope that jurors will be
  convinced that such evidence is extremely
  unlikely if the assumption of innocence were true.

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The Two Types of Errors

• Courtroom Analogy Potential choices and errors

• I. We believe enough evidence to conclude the
  defendant is guilty.
• Potential error An innocent person falsely
  convicted and guilty party remains free.
• usually seen as more serious.
• II. We cannot rule out that defendant is
  innocent, so he or she is set free without
  penalty.
• Potential error A criminal has been erroneously
  freed.

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The Two Types of Errors in Testing

• Type 1 error can only be made if the null
  hypothesis is actually true.
• Type 2 error can only be made if the alternative
  hypothesis is actually true.

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Decision Errors Type I

• If we decide there is a relationship in the
  population (reject null hypothesis)
• This is an incorrect decision only if the null
  hypothesis is true.
• The probability of this incorrect decision is
  equal to the cut-off (?) for the P-value.
• If the null hypothesis is true and the cut-off is
  0.05
• There really is no relationship and the extremity
  of the test statistic is due to chance.
• About 5 of all samples from this population will
  lead us to wrongly reject chance.

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Decision Errors Type II

• If we decide not to reject chance and thus allow
  for the plausibility of the null hypothesis
• This is an incorrect decision only if the
  alternative hypothesis is true.
• The probability of this incorrect decision
  depends on
• the magnitude of the true relationship,
• the sample size,
• the cut-off for the P-value.

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Power of a Test

• This is the probability that the sample we
  collect will lead us to reject the null
  hypothesis when the alternative hypothesis is
  true.
• The power is larger for larger departures of the
  alternative hypothesis from the null hypothesis
  (magnitude of difference).
• The power may be increased by increasing the
  sample size.

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Inference for Population MeansHypothesis Testing

• The last part of this chapter discusses the
  situation when interest is completing hypothesis
  tests about population means rather than
  population proportions.
• DO NOT worry about the details for inference on
  means just try to get the main idea of
  hypothesis testing.

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Test for means
One of the conclusions made by researchers from a
study comparing the amount of bacteria in
carpeted and uncarpeted rooms was, The average
difference in mean bacteria colonies per cubic
foot was 3.48 colonies (95 CI -2.72, 9.68
P-value 0.29). What are the null and
alternative hypotheses being tested here? Is
there a statistically significant difference
between the means of the two groups?
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Answer

• Null The mean number of bacteria for carpeted
  rooms is equal to the mean number of bacteria for
  uncarpeted rooms.
• Alt The mean number of bacteria for carpeted
  rooms is different from the mean number of
  bacteria for uncarpeted rooms.
• P-value is large (gt.05), so there is not a
  significant difference (fail to reject the null
  hypothesis).
• Also, the confidence interval for the
  difference contains 0.

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The Hypotheses for aSingle Mean

• Null H0 m m0
• One-sided alternatives
• Ha m gt m0
• Ha m lt m0
• Two-sided alternative
• Ha m ¹ m0

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The Hypotheses for aDifference in Two Means

• Null H0 mdiff mdiff,0 (usually 0)
• One-sided alternatives
• Ha mdiff gt mdiff,0
• Ha mdiff lt mdiff,0
• Two-sided alternative
• Ha mdiff ¹ mdiff,0

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P-value in one-sided and two-sided tests
One-sided (one-tailed) test
Two-sided (two-tailed) test
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Chapter 23

• Use and Abuse of Statistical Inference

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Question
When presenting the results of a study, would it
be sufficient to only report the P-value? Why
would it be a good idea to also give a confidence
interval based on the results?
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Warnings about Reports on Hypothesis Tests Data
Origins

• For any statistical analysis to be valid, the
  data must come from proper samples. Complex
  formulas and techniques cannot fix bad (biased)
  data. In addition, be sure to use an analysis
  that is appropriate for the type of data
  collected.

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Warnings about Reports on Hypothesis Tests
P-value or C.I.?

• P-values provide information as to whether
  findings are more than just good luck, but
  P-values alone may be misleading or leave out
  valuable information (as seen later in this
  chapter). Confidence intervals provide both the
  estimated values of important parameters and how
  uncertain the estimates are.

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Warnings about Reports on Hypothesis Tests
Significance

• If the word significant is used to try to
  convince you that there is an important effect or
  relationship, determine if the word is being used
  in the usual sense or in the statistical sense
  only.

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Warnings about Reports on Hypothesis Tests Large
Sample

• If a study is based on a very large sample size,
  relationships found to be statistically
  significant may not have much practical
  importance.

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Warnings about Reports on Hypothesis Tests Small
Sample

• If you read no difference or no relationship
  has been found in a study, try to determine the
  sample size used. Unless the sample size was
  large, remember that it could be that there is
  indeed an important relationship in the
  population, but that not enough data were
  collected to detect it. In other words, the test
  could have had very low power.

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Warnings about Reports on Hypothesis Tests 1- or
2- Sided

• Try to determine whether the test was one-sided
  or two-sided. If a test is one-sided, and
  details are not reported, you could be misled
  into thinking there was no difference, when in
  fact there was one in the direction opposite to
  that hypothesized.

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Warnings about Reports on Hypothesis Tests Only
Significant are Reported?

• Sometimes researchers will perform a multitude of
  tests, and the reports will focus on those that
  achieved statistical significance. Remember that
  if nothing interesting is happening and all of
  the null hypotheses tested are true, then about
  1 in 20 (.05) tests should achieve statistical
  significance just by chance. Beware of reports
  where it is evident that many tests were
  conducted, but where results of only one or two
  are presented as significant.

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• Which of the following conclusions should make
  you suspicious as an educated consumer of
  statistical information?
• a. Based on our sample results we know there is
  no relationship between these two variables in
  the population.
• b. We looked at all possible correlations
  between these 10 variables, and this was the only
  one that was significant, signifying its
  tremendous importance.
• c. All of the above.

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Drug Use in American High Schools
Alcohol Use
Bogert, Carroll. Good news on drugs from the
inner city, Newsweek, Feb. 1995, pp. 28-29.
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Drug Use in American High Schools

• Alternative Hypothesis The percentage of high
  school students who used alcohol in 1993 is less
  than the percentage who used alcohol in 1992.
• Null Hypothesis There is no difference in the
  percentage of high school students who used in
  1993 and in 1992.

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Drug Use in American High Schools
1993 survey was based on 17,000 seniors, 15,500
10th graders and 18,500 8th graders.
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Drug Use in American High Schools

• The article suggests that the survey reveals
  good news since the differences are all
  negative.
• The differences are significant.
• statistically?
• practically?

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Quitting Smoking with Nicotine Patches
Compared the smoking cessation rates for smokers
randomly assigned to use a nicotine patch versus
a placebo patch.
Null hypothesis The proportion of smokers in the
population who would quit smoking using a
nicotine patch and a placebo patch are the
same. Alternative hypothesis The proportion of
smokers in the population who would quit smoking
using a nicotine patch is higher than the
proportion who would quit using a placebo patch.
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Quitting Smoking with Nicotine Patches
Higher smoking cessation rates were observed in
the active nicotine patch group at 8 weeks (46.7
vs 20) (P lt .001) and at 1 year (27.5 vs 14.2)
(P .011). (Hurt et al., 1994, p. 595)
Conclusion p-values are quite small less than
0.001 for difference after 8 weeks and equal to
0.011 for difference after a year. Therefore,
rates of quitting are significantly higher using
a nicotine patch than using a placebo patch after
8 weeks and after 1 year.
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Study Proves New Bicycle Seat More Healthy ? FT.
LAUDERDALE, FL - A traditional bicycle seat is
more likely to cause male sexual dysfunction and
a variety of other health problems than a new
seat with a revolutionary design, according to
Medicine and Science In Sports and Exercise, the
official journal of the American College of
Sports Medicine. Results of the 2004 study
concluded that the typical sport/racing saddle
with a narrow protruding nose causes twice the
pressure in the perineal region as the Solution
Bicycle Seat, a unique new model without the
nose. For the study, 33 bicycle police patrol
officers pedaled a stationary bicycle at a
controlled cadence and workload, sitting on a
variety of bicycle seats. Previous evolutions of
the traditional seats were tested, including the
split-horn design with gel-strips said to reduce
shock resistance and make the horn softer. Yet
even these seats did not eliminate the pressure
to the perineal region. The study concluded that
it is the seat design, not the padding that makes
the difference. Source of News Solution Bicycle
Seat
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Effect of bicycle saddle designs on the pressure
to the perineum of the bicyclist. Med Sci Sports
Exerc 2004 Jun Vol. 36 (6), pp.
1055-62. METHODS Saddle, pedal, and handlebar
contact pressure were measured from 33 bicycle
police patrol officers pedaling a stationary
bicycle at a controlled cadence and workload.
Pressure was characterized over the saddle as a
whole and over a region of the saddle assumed to
represent pressure on the cyclist's perineum
located anteriorly to the ischial tuberosities.
RESULTS The traditional sport/racing saddle
was associated with more than two times the
pressure in the perineal region than the saddles
without a protruding nose (P lt 0.01). There were
no significant differences in perineal pressure
among the nontraditional saddles. Measures of
load on the pedals and handlebars indicated no
differences between the traditional saddle and
those without protruding noses. This finding is
contradictory to those studies suggesting a shift
toward greater weight distribution on the
handlebars and pedals when using a saddle without
a nose.
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For the study

• The explanatory variable is pressure and the
  response variable is seat type.
• The explanatory variable is seat type and the
  response variable is pressure.
• The explanatory variable is bicycle police and
  the response variable is erectile dysfunction.

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Given what we know so far, this study was likely
to be

• A sample survey
• An observational study
• An experiment where the same experimental units
  were used for all treatments.

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The news report on the research is suspect
because

• The source of the news is a bicycle seat company
• The extent or the size of the claimed difference
  is not reported.
• The results are extended to a population that may
  not be represented by the sample.
• All of the above.

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The traditional sport/racing saddle was
associated with more than two times the pressure
in the perineal region than the saddles without a
protruding nose (P lt 0.01).

• Given the p-value listed, we know that the null
  hypothesis was
• Rejected.
• Not rejected.

The p-value tells us A) the probability of
getting a sample result as extreme or more
extreme than the one we saw if Ho were
true. B) the probability that Ha is true.
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This study proved beyond a doubt that

• The seat design, not the padding, makes the
  difference.
• Bicycle police patrol officers have a high rate
  of erectile dysfunction.
• Nothing was definitively proven by this one
  study, but the researchers did find a
  relationship between saddle type and perineal
  pressure that is newsworthy for consumers and
  merits further research.

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Key concepts

• Steps of Hypothesis Testing
• P-values and Statistical Significance
• Decision Errors
• Statistically significant vs practically
  important
• Large/Small Samples and Statistical Significance
• Multiple Tests and Statistical Significance

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April 29in classfinal exam (cumulative)

• Next week in class comprehensive review extra
  credit!!