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.

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 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 #2  Medical literature states the mean body temperature of adults is In a random sample of 52 adults, the sample mean body temperature was with a sample standard deviation of Find a 95% confidence interval for the population mean body temperature of adults.

Example #2 (cont.)  d.f. = ___________  t* =  Assumption: ______________________

Example #2 (cont.)

Hypothesis Test for μ  H O : _______________  H A : Three possibilities _____________

Hypothesis Test for μ  Assumptions

Hypothesis Test for μ  Test Statistic

P-value for H A : ___________  P-value = P(t n-1 > t)

P-value for H A : ____________  P-value = P(t n-1 < t)

P-value for H A : _________  P-value = 2*P(t n-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 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.)

 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 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.)

 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.)

 d.f. = ___________  P-value

Example #3 (cont.)  Decision:  Conclusion: