Inferences about Means

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Chapter 23

• Inferences about Means

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Review

• One Quantitative Variable
• Population Mean Value _____
• Population Standard Deviation Value ____

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Review

• Estimate ________
• Take random sample
• Calculate sample mean ________
• Calculate sample standard deviation _______

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Long Term Behavior of Sample Mean Statistic

• Sampling distribution of sample mean
• For variables with normal distributions.
• For variables with non-normal distribution when
  sample size n is large.

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Problem ____________________

• Solution
• Replace _______________ with
• __________________________.
• Standard error of the sample mean

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Sampling distribution of Sample Mean
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The t distribution

• Different t distribution for each value of
  ________.

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Using the t distribution

• Assumptions
• Random sample.
• Independent values.
• No more than 10 of population sampled.
• Nearly Normal Population Distribution.
• __________________________________________

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History of t distribution

• William S. Gosset
• Head brewer at Guinness brewery in Dublin,
  Ireland.
• Field experiments - find better barley and hops.
• Small samples
• Unknown s.
• Published results under name Student.
• t distribution also called Students t.

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The t distribution

• t distribution
• _________________________________
• _________________________________
• _________________________________
• _________________________________

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t distribution table

• Row degrees of freedom.
• Column
• One tail probability.
• Table value t where P(T(n-1) gt t) a
• Two tail probability.
• Table value t where P(T(n-1) gt t) a/2
• t critical value for t distribution.

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Inference for µ

• C Confidence interval for µ.
• t comes from t distribution with (n-1) d.f.

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Example

• Find t for
• 95 CI, n 10
• 90 CI, n 15
• 99 CI, n 25

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Example 1

• A medical study finds that in a sample of 27
  members of a treatment group, the sample mean
  systolic blood pressure was 114.9 with a sample
  standard deviation of 9.3. Find a 90 CI for the
  population mean systolic blood pressure.

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Example 1 (cont.)

• d.f. ___________
• t __________
• Assumption Blood pressure values must have a
  fairly symmetric distribution.

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Example 1 (cont.)
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Example 1 (cont.)
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Example 2

• Medical literature states the mean body
  temperature of adults is 98.6. In a random
  sample of 52 adults, the sample mean body
  temperature was 98.28 with a sample standard
  deviation of 0.68. Find a 95 confidence
  interval for the population mean body temperature
  of adults.

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Example 2 (cont.)

• d.f. ___________
• t 2.009
• Assumption ______________________

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Example 2 (cont.)
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Example 2 (cont.)
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Hypothesis Test for µ

• HO _______________
• HA Three possibilities
• _____________
• _____________
• _____________

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Hypothesis Test for µ

• Assumptions

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Hypothesis Test for µ

• Test Statistic

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P-value for HA ___________

• P-value P(tn-1 gt t)

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P-value for HA ____________

• P-value P(tn-1 lt t)

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P-value for HA _________

• P-value
• 2P(tn-1 gt t)

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Hypothesis Test for µ

• P-value
• Small ________________________________
• _______________________________________
• Large ________________________________
• _______________________________________
• Small and large p-values determined by a.

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Hypothesis Test for µ

• If p-value lt a
• If p-value gt a

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Hypothesis Test for µ

• Conclusion Always stated in terms of problem.

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Example 1

• A medical study finds that in a sample of 27
  members of a treatment group, the sample mean
  systolic blood pressure was 114.9 with a sample
  standard deviation of 9.3. Is this enough
  evidence to conclude that the mean systolic blood
  pressure of the population of people taking this
  treatment is less than 120. Use a 0.1

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Example 1 (cont.)

• Ho____________
• Ha____________
• Assumptions

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Example 1 (cont.)
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Example 1 (cont.)

• d.f. ______________
• P-value

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Example 1 (cont.)

• Decision
• Conclusion

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Example 2

• The manufacturer of a metal TV stand sets a
  standard for the amount of weight the stand must
  hold on average. For a particular type of stand,
  the average is set for 500 pounds. In a random
  sample of 16 stands, the average weight at which
  the stands failed was 490.5 pounds with a
  standard deviation of 10.4 pounds. Is this
  evidence that the stands do not hold the standard
  average weight of 500 pounds? Use a 0.01

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Example 2 (cont.)

• Ho ____________
• Ha ____________
• Assumptions

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Example 2 (cont.)
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Example 2 (cont.)

• d.f. ________
• P-value

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Example 2 (cont.)

• Decision
• Conclusion

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Example 3

• During an angiogram, heart problems can be
  examined through a small tube threaded into the
  heart from a vein in the patients leg. It is
  important the tube is manufactured to have a
  diameter of 2.0mm. In a random sample of 20
  tubes, they find the mean diameter of the tubes
  is 2.01mm with a standard deviation of 0.01mm.
  Is this evidence that the diameter of the tubes
  is different from 2.0mm? Use a 0.01

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Example 3 (cont.)

• Ho______________
• Ha______________
• Assumptions

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Example 3 (cont.)
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Example 3 (cont.)

• d.f. ___________
• P-value

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Example 3 (cont.)

• Decision
• Conclusion