Conditions Required for a Valid Large-Sample Confidence Interval for µ

2. Conditions Required for a Valid Large-SampleConfidence Interval for µ 1. A random sample is selected from the target population. 2. The sample size n is large (i.e., n ≥ 30). Due to the Central Limit Theorem, this condition guarantees that the sampling distribution of is approximately normal. Also, for large n, s will be a good estimator of .

5. Thinking Challenge • Wehave a randomsample of customerordertotalswith an average of $78.25 and a populationstandarddeviation of $22.5. • A) Calculate a 90% confidenceintervalforthemeangiven a sample size of 40 orders. • B) Calculate a 90% confidenceintervalforthemeangiven a sample size of 75 orders. • C) Explainthedifference in the 90% confidenceintervalscalculated in A and B. • D)Calculate the minimum sample size needed to identify a 90% confidence interval for the mean assuming a $5 margin of error.

6. 6.3 Confidence Interval for a Population Mean:Student’s t-Statistic

7. Smallsample size problem forinferenceabout • Theuse of a smallsample in makinginferenceabout presentstwoproblemswhenweattempttousethestandard normal z as a test statistic.

8. Problem 1 • Theshape of thesamplingdistribution of thesamplemeannowdepends on theshape of thepopulationsampled. • We can no longerassumethatthesamplingdistribution of samplemean is approximately normal becausethecentral limit theoremensuresnormalityonlyforsamplesthataresufficientlylarge.

9. Solutionto Problem 1 • Weknowthatifoursamplecomesfrom a populationwith normal distributionthesamplingdistribution of samplemeanwill be normal regardless of thesample size.

10. Problem 2 • Thepopulationstandarddeviation is almostalwaysunknown. Forsmallsamplesthesamplestandarddeviaitonsprovidespoorapproximationfor .

11. Solutionto Problem 2(Small Sample with known) Use the standard normal statistic

12. Solutionto Problem 2(Small Sample with Unknown) Instead of using the standard normal statistic use the t–statistic in which the sample standard deviation, s, replaces the population standard deviation, .

13. Student’s t-Statistic The t-statistic has a sampling distribution very much like that of the z-statistic: mound-shaped, symmetric, with mean 0. The primary difference between the sampling distributions of t and z is that the t-statistic is more variable than the z-statistic.

14. Degrees of Freedom The actual amount of variability in the sampling distribution of t depends on the sample size n. A convenient way of expressing this dependence is to say that the t-statistic has (n – 1) degrees of freedom(df).

15. Student’s t Distribution Standard Normal Bell-Shaped Symmetric ‘Fatter’ Tails t (df = 13) t (df = 5) z t 0 Thesmallerthedegrees of freedomfor t-statistic, themorevariablewill be itssamplingdistribution.

18. Wehave a randomsample of 15 cars of thesame model. Assumethatthegasmilageforthepopulation is normallydistributedwith a standarddeviaition of 5.2 milesper galon. • A) Identifytheboundsfor a 90% confidenceintervalforthemeangiven a samplemean of 22.8 milespergallon. • B) The car manufacturer of thisparticular model claimsthattheaveragegasmilage is 26 milespergallon. Discussthevalidity of thisclaimusingthe 90% confidenceintervalcalculated in A. • C) Let a and b representthelowerandupperboundaries of 90% confidenceintervlforthemean of thepopulation. Is it correcttoconcludethat tere is a 90% probabilitythattruepopulationmeanliesbetween a and b?

19. Thinking Challenge • In 1882 Michelsonmeasuredthespeed of light. His values in km/secand 299,000 substractedfromthem. He reportedtheresults of 23 trialswith a mean of 756.22 and a standarddeviaition of 107.12. • Find a 95% confidenceintervalforthetruespped of lightfromthesestatistics. • Interpretyourresult.