Confidence Interval and Hypothesis Testing for:

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Confidence Interval and Hypothesis Testing for

• Population Mean ( ?)

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Assumptions Conditions

• Random sample
• Independent observations
• Nearly normal distribution
• y N (?, ?/ n )

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Students t-Model for decisions about the mean, ?
-
y - ?
t
s
n
With dfn-1
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One-Sample t-Interval

• When the conditions are met, the confidence
  interval for the means of one population is
  
• _______________________
• where the standard error of the means is
• _______________________
• The critical value (t) depends on the
  particular confidence level, C, and the degrees
  of freedom, df.

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CI for the mean, ?
Stat, tests, 8 T-Interval
HT for the mean, ?
Stat, tests, 2 T-Test
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Terms
Significant Level (?) P-value (P in TI) Null
Hypothesis (Ho ) Alternative Hypothesis (HA )
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Decisions

• Reject the null hypothesis if the P-value is
  less than or equal to the significance level ?.
• Reject Ho if P-value lt ?.
• Fail to reject Ho if P-value gt ?.

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Procedures

• HypothesesHo HA
• 2. Assumptions and Conditions
• 3. Mechanics
• T P-value lt Significant Level (?)?
• 4. Conclusion Answer the original question.

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Confidence Interval and Hypothesis Testing

• Comparing Two Population Means
• Finding and Testing their difference
• (?1- ?2)

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Assumptions and Conditions for t-model

• Independence Assumption (Each condition needs to
  be checked for both groups.)
• Randomization Condition Were the data collected
  with suitable randomization (representative
  random samples or a randomized experiment)?
• 10 Condition Is the sample size (n) less than
  10 of the population size (N)? We dont usually
  check this condition for differences of means. We
  will check it for means only if we have a very
  small population or an extremely large sample.

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Assumptions and Conditions (cont.)

• Normal Population Assumption
• Nearly Normal Condition This must be checked for
  both groups. A violation by either one violates
  the condition.
• Independent Groups Assumption The two groups we
  are comparing must be independent of each other.

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Two-Sample t-Interval

• When the conditions are met, the confidence
  interval for the difference (between means of two
  independent groups) is
  
• where the standard error of the difference of
  the means is
• The critical value (t) depends on the
  particular confidence level, C, and the degrees
  of freedom, df, derived from the sample sizes and
  a special formula.

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Degrees of Freedom (df)

• The special formula for the degrees of freedom
  for our t critical value is a bear
• Because of this, we will let technology calculate
  degrees of freedom for us!
• (or pursue a stat major or minor)

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Testing the Difference Between Two Means

• The hypothesis test we use is the
• two-sample t-test for means.
• The conditions for the two-sample t-test for the
  difference between the means of two independent
  groups are the same as for the two-sample
  t-interval.

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Testing the Difference Between Two Means (cont.)

• We test the hypothesis H0?1 ?2 ?0, where
  the hypothesized difference, ?0, is almost always
  0, using the statistic
• The standard error is
• When the conditions are met and the null
  hypothesis is true, this statistic can be closely
  modeled by a Students t-model with a number of
  degrees of freedom given by a special formula. We
  use that model to obtain a P-value.

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Back Into the Pool

• Remember that when we know a proportion, we know
  its standard deviation.
• Thus, when testing the null hypothesis that two
  proportions were equal, we could assume their
  variances were equal as well.
• This led us to pool our data for the hypothesis
  test for p1-p2.

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Back Into the Pool (cont.)

• For means, there is also a pooled t-test.
• Like the two-proportions z-test, this test
  assumes that the variances in the two groups are
  equal.
• But, be careful, there is no link between a mean
  and its standard deviation

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Back Into the Pool (cont.)

• If we are willing to assume that the variances of
  two means are equal, we can pool the data from
  two groups to estimate the common variance and
  make the degrees of freedom formula much simpler.
• We are still estimating the pooled standard
  deviation from the data, so we use Students
  t-model, and the test is called a pooled t-test.

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The Pooled t-Test

• If we assume that the variances are equal, we can
  estimate the common variance from the numbers we
  already have
• Substituting into our standard error formula, we
  get
• Our degrees of freedom are now df n1 n2 2.

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The Pooled t-Test and Confidence Interval

• The conditions are the same, plus the assumption
  that the variances of the two groups are the
  same.
• For the hypothesis test, our test statistic is
• which has df n1 n2 2.
• Our confidence interval is

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Is the Pool All Wet?

• So, when should you use pooled-t methods rather
  than two-sample t methods? Well, hardly ever.
• Because the advantages of pooling are small, and
  you are allowed to pool only rarely (when the
  equal variance assumption is met).
• Dont pool.

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Can We Test Whether the Variances Are Equal?

• The test is very sensitive to non-normal data and
  works poorly for small sample sizes.
• So, the test does not work when we need it to.

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What Can Go Wrong?

• Watch out for paired data.
• The Independent Groups Assumption deserves
  special attention.
• If the samples are not independent, you cant use
  two-sample methods.
• Look at the plots.
• Check for outliers and non-normal distributions
  by making and examining boxplots or normal
  probability plots.

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What have we learned?

• To use statistical inference to compare the
  means of two independent groups.
• We use t-models for CI and HT.
• It is important to check conditions to see if the
  assumptions for t-model are met.
• Dont pool the standard errors.

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Coke Versus Pepsi

• Independent random samples of 36 cans of Coke
  and Pepsi are weighed and summarized below. Use
  the 0.01 significance level to test the claim
  that the mean weight of regular Coke is different
  from the mean weight of regular Pepsi.
• Regular Coke Regular Pepsi
• n 36 36
• y 0.817 0.824
• s 0.0076 0.0057

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Coke Versus Pepsi
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Coke vs. Pepsi

• Ho ?1 ? 2
• Ha ? 1 ? ? 2
• 0.01

Fail to reject H0
Reject H0
Reject H0
- t - ____
t _____
t 0