chapter 23 inferences about means. review one quantitative variable population mean value _____ ...
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Chapter 23
Inferences about Means
Review One Quantitative Variable
Population Mean Value _____
Population Standard Deviation Value ____
Review Estimate ________
Take random sample Calculate sample mean ________ Calculate sample standard deviation _______
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.
Problem: ____________________ Solution:
Replace _______________ with __________________________.
Standard error of the sample mean
Sampling distribution of Sample Mean
The t distribution
Different t distribution for each value of ________.
Using the t distribution Assumptions
Random sample. Independent values. No more than 10% of population sampled. Nearly Normal Population Distribution.
__________________________________________
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 σ.
Published results under name Student. t distribution also called Student’s t.
The t distribution t distribution
_________________________________ _________________________________ _________________________________ _________________________________
t distribution table Row = degrees of freedom. Column
One tail probability. Table value = t* where P(T(n-1) > t*) = α
Two tail probability. Table value = t* where P(T(n-1) > t*) = α/2
t* = critical value for t distribution.
Inference for μ C% Confidence interval for μ.
t* comes from t distribution with (n-1) d.f.
n
sty *
Example Find t* for
95% CI, n = 10
90% CI, n = 15
99% CI, n = 25
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.
Example #1 (cont.) d.f. = ___________
t* = __________
Assumption: Blood pressure values must have a fairly symmetric distribution.
Example #1 (cont.)
Example #1 (cont.)
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.
Example #2 (cont.) d.f. = ___________
t* = 2.009
Assumption: ______________________
Example #2 (cont.)
Example #2 (cont.)
Hypothesis Test for μ HO: _______________
HA: Three possibilities _____________ _____________ _____________
Hypothesis Test for μ Assumptions
Hypothesis Test for μ Test Statistic
P-value for HA: ___________ P-value = P(tn-1 > t)
P-value for HA: ____________ P-value = P(tn-1 < t)
P-value for HA: _________ P-value =
2*P(tn-1 > |t|)
Hypothesis Test for μ P-value
Small _______________________________________________________________________
Large _______________________________________________________________________
Small and large p-values determined by α.
Hypothesis Test for μ If p-value < α
If p-value > α
Hypothesis Test for μ Conclusion: Always stated in terms of
problem.
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 α = 0.1
Example #1 (cont.) Ho:____________ Ha:____________
Assumptions
Example #1 (cont.)
Example #1 (cont.) d.f. = ______________
P-value
Example #1 (cont.) Decision:
Conclusion:
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 α = 0.01
Example #2 (cont.) Ho: ____________ Ha: ____________
Assumptions
Example #2 (cont.)
Example #2 (cont.) d.f. = ________
P-value
Example #2 (cont.) Decision:
Conclusion:
Example #3 During an angiogram, heart problems can be
examined through a small tube threaded into the heart from a vein in the patient’s 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 α = 0.01
Example #3 (cont.) Ho:______________ Ha:______________
Assumptions
Example #3 (cont.)
Example #3 (cont.) d.f. = ___________
P-value
Example #3 (cont.) Decision:
Conclusion: