1. Populations and Samples Hypothesis Testing Example

2. Lecture Objectives You should be able to: 1.Set up the Null and Alternate Hypotheses 2.Conduct a one-sample Hypothesis Test for the Mean. 3.Interpret the results

3. The Situation You believe that the current price of unleaded regular gasoline is less than $4.00 on average nationwide, and wish to prove it. Set up the hypothesis and test it.

4. Null and Alternate Hypotheses What we wish to prove is called the Alternate Hypothesis. The opposite of that is the Null, which must be assumed and shown to be unlikely, based on sample data. H 0 : μ = 4.00 H a : μ < 4.00

5. What constitutes proof ? Any conclusion based on a sample may be wrong. What probability (at most) of being wrong is acceptable to you? This is called (alpha), or the acceptable Type I Error. Let = 0.05 (or 5%)

6. The Sample Data A sample of 49 gas stations nationwide shows average price of unleaded is $ 3.87 and a standard deviation of $ 0.15. Could this sample have come from a population where the Mean was in fact $4.00 (or greater)? Assume the null is true, and this sample did in fact come from such a population.

7. Sampling Distribution if H 0 True What would the distribution of sample means from such a population look like? From the Central Limit Theorem, we have the following: = $4.00 = = 0.15/ √ 49 = $ 0.02143

8. The Test Statistic How far from the assumed mean of 4.00 is the observed sample mean of 3.87? Measured in Standard Errors, this is the t-statistic. t = (3.87- 4.00)/0.02143 = -6.06 How likely is it that a sample mean would be this far away (or farther) from the population mean?

9. p-value The probability that a value would be as extreme as (or more extreme than) 6.06 SEs below the Mean is: 0.0000001! [In Excel, =TDIST(6.06,48,1)] This is called the p-value of the Hypothesis test.

10. Conclusion If the null were true (the average price were in fact 4.00), there is only a 0.0000001 probability that you would pick a sample with a mean of 3.87 or smaller from such a population. Still, you did pick it! Therefore, either the null must be false (and therefore you proved your case) or you picked an extremely rare sample.

11. Conclusion (2) You can conclude that the sample could not have come from a population with Mean = 4.00 as assumed, and instead must have come from one with Mean < 4.00. The chance that you are wrong is less than 5%, your tolerance level. In other words, p <, hence you proved the case beyond reasonable doubt.