Statistical Inferences Based on Two Samples

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Statistical Inferences Based on Two Samples

• 9.2 Comparing Two Population Means Using Small
  Independent Samples and Assuming Sigmas are
  Unknown

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Sampling Distribution of
Normal, if each of the sampled populations is
normal and approximately normal if the sample
sizes n1 and n2 are large
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Sampling Distribution of
(Continued)
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Large Sample Confidence Interval, Difference in
Mean
If two independent samples are from populations
that are normal or each of the sample sizes is
large, 100(1 - a) confidence interval for m1 -
m2 is
If ?1 and ?2 are unknown estimate the sample
standard deviations by s1 and s2 and use the t
distribution
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9.2 Comparing Two Population Means Using
Independent Samples with Sigmas Unknown
If two independent samples are from populations
that are normal with equal variances, 100(1 - a)
confidence interval for m1 - m2 is
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Tests about Differences in Means When Variances
are Equal
If sampled populations are both normal with equal
variances, we can reject H0 ?1 - ?2 D0 at the
? level of significance if and only if the
appropriate rejection point condition holds or,
equivalently, if the p-value is less than ?.
Reject H0 if
p-Value
Alternative
t?, t?/2 and p-values are based on (n1 n2 2)
df
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Example Difference in Mean Test, Assuming Equal
Variances
Catalyst Case, Difference in Mean Hourly Yields?
Test H0 ?1 - ?2 0 versus Ha ?1 - ?2 ? 0, ?
0.01
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Hypothesis Test and Confidence Interval Example
2 Exercise 9.21, pg. 359

• What are we given? n1 22 s1 225 xbar1
  1500 n2 22 s2 251 xbar2 1300 ? .05
• First assume equal population variances
• Step 1, establish hypotheses
• H0 ?1 - ?2 0 vs. Ha ?1 - ?2 gt 0
• Step 2, set significance level. a .05 (given)
• Step 3, compute the test statistic, but first the
  pooled variance

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Hypothesis Test and Confidence Interval Example
2 Exercise 9.21, pg. 359

• Step 4a, determine the rejection point, t.05,42
  1.684
• Step 4b, estimate the p-value. Using df 42,
  t-table gives P(T gt 3.307) .001 and P(T gt
  2.704) .005 Since 2.704 lt (t 2.78) lt 3.307,
  p-value is between 0.001 and 0.005
• Step 5, decision reject Ho since (a) test
  statistic, t (2.78) gt rejection point (1.684) or
  (b) p-value (between .001 .005) lt ? .05

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Using df 42, t-table gives P(T gt 2.704) .005
and P(T gt 3.307) .001. With t 2.78,
p-value is between .001 and .005
0.005
0.001
t 2.704 2.78
3.307
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Hypothesis Test Example 2

• Step 6, conclusion within context there is very
  strong evidence that type A training results in
  higher mean weekly sales than does type training.
  

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MegaStat Output for Example 2
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100(1 - a) confidence interval for m1 - m2 is
given by
We are 95 confident that mean weekly sales with
type A train-ing exceeds that with type B
training by between 42 and 358