Module 4 Hypothesis Testing for One, Two and Three Samples

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Module 4Hypothesis Testing for One, Two and
Three Samples

• Chapter 8, 9, 10

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One sample hypothesis testing
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Sample Means Curve
My sample mean Is it part of the null hypothesis
population?
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Sample Means Curve
My sample mean Is it part of a population where
the mean is less than mu?
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Chapter 8 - Hypothesis Testing

• When you do hypothesis testing you are answering
  the question Have I found anything different?
• Do mammograms reduce the risk of dying from
  breast cancer? Is having a mammogram different
  from not having a mammogram in preventing death
  from breast cancer?
• Swedish study 63 of 21,088 who had mammograms
  died. 66 of 21,195 who did not have mammograms
  died.

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Chapter 8 - One Sample Hypothesis Testing

• Translate your study to numerical values
  (mortality percent who die)
• State your alternative hypothesis first (what
  youre trying to prove)
• State your null hypothesis (status quo)

Mammograms reduce mortality. Mammogram lt Not
having a mammogram
Mammograms do not reduce mortality. Mammgram gt
Not having a mammogram
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Chapter 8 - One Sample Hypothesis Testing

• The court case

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Chapter 8 - One Sample Hypothesis Testing

• Remember samples can sit anywhere on a
  distribution. How do you decide when you have
  something different?

Each type of error will have its costs. Type 1
more likely when alpha is bigger. Type 2 more
likely when alpha is smaller.
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Choosing the right alpha
Large alpha means that there is a higher
likelihood of rejecting Ho or TYPE 1 error
rejecting Ho when it is TRUE
Small alpha means that there is a higher
likelihood of NOT rejecting Ho or TYPE 2 error
rejecting Ha when it is TRUE
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Chapter 8 - One Sample Hypothesis Testing
Language Hints
Two Tail Equal Exactly the same Not changed Same
as Compare Is different
One Tail - Upper or (lower) Is greater
(less) More (below) Larger (smaller) Longer
(shorter) Higher (lower) Better (reduced) At
least (Not more) Not less than (at most)
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Chapter 8 - One Sample Hypothesis Testing

• Specify alternative and null hypothesis in
  numerical values. Select your significance
  (alpha) level.
• One tail greater than (alpha in one tail)
• One tail less than (alpha in one tail)
• Two tail equal/does not equal (alpha divided into
  two tails)
• Take a random sample of the population of
  interest.
• Determine the test statistic. (For this chapter,
  either z for ngt30 or t for nlt30)
• Specify the rejection area based on your
  significance level. Express as a critical z or
  critical t.
• Confront your test z with your critical z. Come
  to a conclusion
• Reject the null hypothesis
• Do not reject the null hypothesis

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Chapter 8 - One Sample Hypothesis Testing
When ngt30
Does your test z fall in the rejection area? If
yes, reject the null hypothesis.
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Chapter 8 - One Sample Hypothesis Testing
REJECT Ho IF
P-value is the probability in the tail associated
with the test statistic. To find it, look up the
test z and find the probability associated with
it in the body of the z table and subtract from
0.5. YOU REJECT THE NULL HYPOTHESIS IF YOUR
P-VALUE IS SMALLER THAN YOUR ALPHA OR
SIGNIFICANCE LEVEL.
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Chapter 8 - One Sample Hypothesis Testing
P-value for two tail is double the probability
for the tail of the test statistic. Find the
probability in the z table and subtract from 0.5
(tail) then multiply by two.
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Chapter 8 - One Sample Hypothesis Testing

• What does a p-value give you that a critical z
  test doesnt?

Tells you the magnitude of your rejection or
nonrejection. How close were you to coming to a
different conclusion? Should you reconsider
methodology or significance level?
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Chapter 8 - One Sample Hypothesis Testing

• When n lt 30 and the underlying population is
  normal, use the t-table to look up the p-value.
• Look up the degrees of freedom or n - 1 and find
  the closest t-value to your test t.

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Chapter 8 - One Sample Hypothesis Testing
n lt 30 Underlying population is normal
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To find P-value on a t distribution, scan the
body of the t table for the closest t value to
your test t. (Example test t3.0) Use the same
df. (n20 df19) Look up p-value at the top of
table. (.001ltp-valuelt.005) Use Excel or TI for
more accuracy.
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Chapter 8 - One Sample Hypothesis Testing
Null hypothesis proportion
Sample proportion
Number in sample
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Chapter 9 - Comparing Two Populations
Question Are two populations different?
Translate to Is there a significant difference
between the means of the two populations?
Standard error
Standard error of the difference
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Comparing two samples curves
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Are the means different?
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Chapter 9 - Comparing Two Populations
Large sample means case
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Chapter 9 - Comparing two populations
Large sample means confidence interval Confidence
interval of the difference between two
population proportions can be calculated using
the standard error of the difference and
confidence level just as we did for a single
population confidence interval.
Confidence level associated with 1-alpha
Standard error of the difference
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Chapter 9 - Comparing Two Populations
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Chapter 9 - Comparing Two PopulationsConfidence
interval of the differenceSmall sample pooled
variance
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Chapter 9 - Comparing Two Populations
Matched Pairs Small or large sample
Used when have dependent sampling, i.e., matched
pair or before/after. Calculate the difference
for each pair. Treat the difference like a
single x value. Calculate mean and standard
deviation. Null hypothesis difference can be
nonzero.
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Chapter 9 - Comparing Two Populations
Difference between two population proportions.
Use combined proportion for standard error of the
difference when the null hypothesis is zero.
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Chapter 9 - Comparing two populations
Confidence interval of the difference in
population proportions can be calculated using
the standard error. Note this standard error is
not combined.
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Chapter 9 Determining sample size
Sample size for estimating population mean
Sample size for estimating population proportion
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Are the means different ANOVA for three or more
samples
ALTERNATE HYPOTHESIS
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ANOVA

• When you have three or more samples, you use
  Analysis of Variance as your test to see if the
  means are different
• F-test with p-value
• Same concept as two-sample pooled variance test
  (F is t squared)

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Chapter 10 - ANOVA
May want to assess the effect of different
factors on responses Devise a analysis of
variance experiment For example the effect of
three different sales promotions in two stores on
weekly sales Weekly sales (the response) will be
measured The three promotions are the
factors The two stores are the level Will compare
six treatments
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Chapter 10 - ANOVA

• Sum of the Squares of Treatment
• Sum of Squares for Error
• Mean Square Treatment
• Mean Square Error
• F

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Chapter 10 - ANOVA
Null hypothesis is that there is no difference
between the means. Alternate hypothesis is that
at least two means differ. Use the F statistic as
your test statistic. It tests the between-sample
variance (difference between the means) against
the within-sample variance (variability within
the sample). The larger this is the more likely
the means are different. Degrees of freedom for
numerator is k-1 (k is the number of
treatments) Degrees of freedom for the
denominator is n-k (n is the number of
responses) If test F is larger than critical F,
then reject the null. If p-value is less than
alpha, then reject the null.
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Chapter 10 - ANOVA
Assumptions All k population probability
distributions are normal. The k population
variances are equal. The samples from each
population are random and independent.
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Chapter 10 - ANOVA
WHEN YOU REJECT THE NULL For an one-way ANOVA
after you have rejected the null, you may want to
determine which treatment yielded the best
results. Must do follow-on analysis to determine
if the difference between each pair of means if
significant. For two-way ANOVA, need to test for
interaction effect. If interaction between
factors then ANOVA is not valid.