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Chapter 2: Comparing Means
2.1 One-Sample t-Test
2.2 Paired t-Test
2.3 Two-Sample t-Test
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Chapter 2: Comparing Means
2.1 One-Sample 2.1 One-Sample tt-Test-Test
2.2 Paired t-Test
2.3 Two-Sample t-Test
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Objectives Use JMP to evaluate the t-test assumption
of normality. Construct lower-tailed, upper-tailed, and two-tailed
hypothesis tests using the t-test. Identify the appropriate test statistic and hypothesis
test to answer a research question.
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One-Sample t-Test AssumptionsThe one-sample t-test assumes that is normally distributed. Normal populations yield normal sample means.
– Decide whether population is normal based on tests of sample data.
– Small samples might weakly support these tests. Non-normal populations might yield normal sample
means.– Large samples benefit from principle of central
limit theorem.
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y
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Evaluating NormalityIn addition to graphical tools, JMP provides a statistical test for normality with the following hypotheses:
H0: the data come from a
normal distribution
H1: the data do not come from
a normal distribution
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Statistical Tests for NormalityThree tests are provided by the Distribution platform to assess the normality of the data. Shapiro-Wilk test for up to 2000 observations. Kolmogorov-Smirnov-Lilliefors (KSL) test for more
than 2000 observations. Lilliefors confidence region in the normal quantile plot.
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One-Sample t-Test AssumptionsThe one-sample t-test assumes that observations are independent. Independence is determined by the data collection
method. An appropriate simple random sample should lead
to independent observations. Advanced methods handle restrictions to the
randomization both in the design and in the analysis of data from an experiment.
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Three Research Problems An electronics manufacturer wants to determine
whether, on average, its power supplies are supplying significantly more or less than the desired 12 amperes.
An agricultural company is confident that its new fertilizer results in crop yields of more than 55 bushels per acre, on average.
A laboratory group is concerned that the purity of a chemical that it is using is less than 99.4% on average, as the supplier claims.
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How to Choose the Appropriate t-TestExamine the alternative hypothesis to determine what t-test is appropriate. Use a two-tailed test if you suspect the true mean
is different than claimed (H1: 12). Use an upper-tailed test if you suspect that the true
mean is higher than claimed (H1: > 55 ). Use a lower-tailed test if you suspect that the true
mean is lower than claimed (H1: < 99.4).
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2.01 Multiple Answer PollThe null hypothesis is which of the following? (Select all of the correct answers.)
a. The logical opposite of the alternative hypothesisb. The claim that you suspect is truec. Always assumed to be trued. Your instinct
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2.01 Multiple Answer Poll – Correct AnswersThe null hypothesis is which of the following? (Select all of the correct answers.)
a. The logical opposite of the alternative hypothesisb. The claim that you suspect is truec. Always assumed to be trued. Your instinct
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Three t-Tests and Their p-Values
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Two-tailed ( |t| )
Upper-tailed ( > )
Lower-tailed ( < )
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2.02 Multiple Choice PollA professor suspects her class is performing below the department average of 73%. She decides to test this claim. Which of the following is the correct alternative hypothesis?
a. μ < .73
b. μ > .73
c. μ ≠ .73
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2.02 Multiple Choice Poll – Correct AnswerA professor suspects her class is performing below the department average of 73%. She decides to test this claim. Which of the following is the correct alternative hypothesis?
a. μ < .73
b. μ > .73
c. μ ≠ .73
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The Test StatisticThe statistic follows a t-distribution with mean zero and a standard deviation that depends on the degrees of freedom.
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where µ0 is the hypothesized mean.
y is the sample mean.
ys is the estimated standard error
of the mean.
ysy
t)μ( 0
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Research QuestionA research company suspects that drivers are going faster than the posted speed limit of 45 mph. Help it test the following hypotheses:
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H0: ≤ 45 H1: > 45
= 0.05
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The Reference Distribution
Compute
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ysy
t)45(
for your sample and compare the sample t to the reference distribution.
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Analysis WorkflowFollow these steps in your analysis: Plot your data Diagnose data problems Perform your analysis / Choose your model Evaluate model assumptions Diagnose statistics or model problems Iterate as necessary
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This demonstration illustrates the concepts discussed previously.
Upper-Tailed Hypothesis Test
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Setup for the PollExamine the Goodness-of-Fit Test output from the demonstration.
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2.03 PollAt α=.05, is there sufficient evidence to reject normality of the data?
Yes
No
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2.03 Poll – Correct AnswerAt α=.05, is there sufficient evidence to reject normality of the data?
Yes
No
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This exercise reinforces the concepts discussed previously.
Exercise
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2.04 Multiple Choice PollIn the exercise, a one-sample t-test examined whether the average weight of cereal boxes was significantly different from 15. What type of test was run?
a. An upper-tailed testb. A lower-tailed testc. A two-sided test
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2.04 Multiple Choice Poll – Correct AnswerIn the exercise, a one-sample t-test examined whether the average weight of cereal boxes was significantly different from 15. What type of test was run?
a. An upper-tailed testb. A lower-tailed testc. A two-sided test
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Chapter 2: Comparing Means
2.1 One-Sample t-Test
2.2 Paired 2.2 Paired tt-Test-Test
2.3 Two-Sample t-Test
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Objectives Recognize when a paired t-test is appropriate. Perform a paired t-test.
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Paired t-TestA paired t-test is a one-sample t-test on the mean of the differences between paired observations. A paired t-test tests the following hypotheses:
where d is the difference between paired measurements.
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H0: µd = 0
H1: µd 0
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The Test StatisticUse this test statistic to determine whether two means from paired samples differ:
where
is the mean difference between the paired observations (di = y2i – y1i).
is the hypothesized mean difference (H0: µd = 0).
is the standard error of the mean difference.
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dddd s
ds
ds
ddt 00
d
ds0d
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Paired t-Test AssumptionsThe assumptions of the paired t-test are as follows: is normally distributed measurements among subjects are independent the difference should not be proportional to the
response
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d
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Total Cholesterol ExampleA clinical chemist is interested in whether cholesterol measurements are different for pairs of fresh and frozen serum samples obtained from a group of patients.
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This demonstration illustrates the concepts discussed previously.
Paired t-Test
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2.05 Multiple Answer PollWhich of the following statements are true?
a. A paired t-test is the same as a one-sample t-test on the differences.
b. The Matched Pairs platform in JMP allows an evaluation of the relationship, if any, between the differences and the response level.
c. A paired t-test is used when pairs of observations are related.
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2.05 Multiple Answer Poll – Correct AnswersWhich of the following statements are true?
a. A paired t-test is the same as a one-sample t-test on the differences.
b. The Matched Pairs platform in JMP allows an evaluation of the relationship, if any, between the differences and the response level.
c. A paired t-test is used when pairs of observations are related.
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This exercise reinforces the concepts discussed previously.
Exercise
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2.06 QuizExamine the output from the exercise. What is the conclusion from these comparisons of blood pressure before and after the training?
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2.06 Quiz – Correct AnswerExamine the output from the exercise. What is the conclusion from these comparisons of blood pressure before and after the training?
The 95% confidence interval on the true average does not contain zero in either case, so this indicates that there is a difference, on average, between the blood pressure readings before and after the training.
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Chapter 2: Comparing Means
2.1 One-Sample t-Test
2.2 Paired t-Test
2.3 Two-Sample 2.3 Two-Sample tt-Test-Test
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Objectives Distinguish between a two-sample t-test
and a paired t-test. Identify the three assumptions of a two-sample t-test. Compare the means of two populations using
a two-sample t-test.
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Two-Sample t-Test or Paired t-Test? A two-sample t-test, or independent t-test, is used
when you compare the means of two independent populations.
A paired t-test is used when there are pairs of related measurements.
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2.07 Multiple Choice PollSuppose that you are interested in evaluating the effect of a marketing campaign on regional retail sales. You collect data before and after the campaign for each retail facility. Which is the more appropriate test?
a. One-sample t-testb. Paired t-testc. Two-sample t-testd. Any of these tests could be appropriate.
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2.07 Multiple Choice Poll – Correct AnswerSuppose that you are interested in evaluating the effect of a marketing campaign on regional retail sales. You collect data before and after the campaign for each retail facility. Which is the more appropriate test?
a. One-sample t-testb. Paired t-testc. Two-sample t-testd. Any of these tests could be appropriate.
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Concrete Example
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Reinforced Standard
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Statistical Assumptions Independent observations is normally distributed for each population Equal variances for each population
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y
C o m p a r in g Tw o P o p u la t io n s
A BTr e a tm e n t
1
2
µ1
µ2
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The Test StatisticUse this t-test to test whether the means of two independent groups are different:
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ty ysy y
y y( )
( )1 2
1 2
1 2
where
)2y1y( t is the test statistic.
iy is the mean for the sample from population i (i=1, 2).
)2y1y( s is the pooled standard error.
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Example of a Two-Sided HypothesisSuppose you are interested in determining whether two group means are different from one another. The null and alternative hypotheses are the following:
H0: the group means are identical (1 = 2)
H1: the group means are different (1 2)
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This demonstration illustrates the concepts discussed previously.
Comparing Two Means Using a t-Test
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Setup for the PollExamine the output of the tests of equal variances.
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2.08 PollAt α =.05, is there sufficient evidence to reject equal variances?
Hint: recall that H0: σ12 = σ2
2 and H1: σ12 ≠ σ2
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Yes
No
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2.08 Poll – Correct AnswerAt α =.05, is there sufficient evidence to reject equal variances?
Hint: recall that H0: σ12 = σ2
2 and H1: σ12 ≠ σ2
2
Yes
No
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This exercise reinforces the concepts discussed previously.
Exercise
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Setup for the PollExamine the output from the exercise to determine which p-value would be appropriate if someone were to test whether the risks at public hospitals were lower than private hospitals.
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2.09 Multiple Choice PollWhich p-value would be appropriate if someone were to test if the risks at public hospitals were lower than private hospitals?
a. 0.4411b. 0.2206c. 0.7794
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2.09 Multiple Choice Poll – Correct AnswerWhich p-value would be appropriate if someone were to test if the risks at public hospitals were lower than private hospitals?
a. 0.4411b. 0.2206c. 0.7794
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