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: