Chapter 8

1. Hypothesis Test Chapter 8

2. Steps to a Hypothesis Test • Hypotheses • Null Hypothesis (Ho) • Alternative Hypothesis (Ha) • Alpha • Distribution (aka model) • Test Statistics and P-value • Decision • Conclusion

3. Steps to a Hypothesis Test • Can remember the steps by the sentence: “Happy Aunts Make The Darndest Cookies”

4. Example 1– Hypothesis Testing • An attorney claims that more than 25% of all lawyers advertise. A sample of 200 lawyers in a certain city showed that 63 had used some form of advertising. At α = 0.05, is there enough evidence to support the attorney’s claim?

5. Hypotheses (Sets up the two sides of the test) • Build the Alternative Hypothesis (Ha) first. • based on the claim you are testing (you get this from the words in the problem) • Three choices • Ha: parameter ≠ hypothesized value • Ha: parameter < hypothesized value • Ha: parameter > hypothesized value • Build Null Hypothesis (Ho) next. • opposite of the Ha (i.e. = , ≥ , ≤ )

6. Example 1– Constructing Hypotheses • We need to know what parameter we are testing and which of the three choices for alternative hypothesis we are going to use. • “An attorney claims that more than 25% of all lawyers advertise” tells us that this is a test for proportions so our parameter is p. • “claims that more than 25%” tells us that Ha: p > .25 and therefore Ho: p ≤ .25

7. Alpha • Alpha = α = significance level • How much proof we are requiring in order to reject the null hypothesis. • The complement of the confidence level that we learned in the last chapter • Usually given to you in the problem, if not, you can choose. • Most popular alphas: 0.05, 0.01, and 0.10

8. Example 1 – Alpha • “At α = 0.05” is given to us in the problem so we just copy α = 0.05

9. Model • The model is the distribution used for the parameter that you are testing. These are just the same as we used in the confidence intervals. • p and μ (n ≥ 30) use the normal distribution • μ (n < 30) uses the t-distribution • uses the chi-squared distribution

10. Example 1 - Model • The model used for a proportion is the normal.

11. Test Statistic • You will have a different test statistic for each of the four different parameters that we have learned about. • p : • μ (n ≥ 30) :

12. Test Statistic • You will have a different test statistic for each of the four different parameters that we have learned about. • μ (n < 30) : • :

13. p-value • This is the evidence (probability) that you will get off of your chart and then compare against your criteria (alpha). • You will need to find the appropriate probability that goes with your Ha. • > and < Ha’s are called one-tailed tests. • ≠ Ha’s are called two-tailed tests. • For z and χ2 you have to take the > probability X2

14. Example 1 – Test Statistic and p-value • The formula for a test statistic for proportions is: • So, from our problem we need a proportion from a sample (p-hat), the proportion from our hypothesis (po), and a sample size (n).

15. Example 1 – Test Statistic and p-value • “A sample of 200 lawyers in a certain city showed that 63 had used some form of advertising” tells us that • p-hat = 63/200 or 0.315 • From our hypothesis we know • po = 0.25 (which means that qo = 0.75) • “sample of 200” tells us that • n = 200

16. Example 1 – Test Statistic and p-value • So our test statistic and p-value are

17. Decision – (always about Ho) • We have two choices for decision • Reject Ho • Do Not Reject Ho • If our evidence (p-value) is less than α we REJECT Ho. • If our evidence (p-value) is greater than α we DO NOT REJECT Ho.

18. Example 1 - Decision • Our p-value is 0.0170 and our alpha is 0.05 • So, since our p-value is less than our alpha our decision is: REJECT Ho.

19. Conclusion – (always in terms of Ha) • Conclusions • Reject Ho • “There is enough evidence to suggest (Ha).” • Do Not Reject • “There is not enough evidence to suggest (Ha).”

20. Example 1 - Conclusion • Our decision to was to reject Ho, so our conclusion is: “There is enough evidence to suggest that p>0.25”

21. Example 1 - Summary • Ho: p ≤ 0.25 Ha: p > 0.25 • α = 0.05 • Model: Normal • z = 2.12 and p-value = 0.0170 • Reject Ho • There is enough evidence to suggest that p>0.25.

22. Example 2 – Hypothesis Testing A researcher reports that the average salary of assistant professors is more than $42,000. A sample of 30 assistant professors has a mean of $43,260. At α = 0.05, test the claim that assistant professors earn more than $42,000 a year. The standard deviation of the population is $5230.

23. Example 2 (cont.) • Hypotheses • Ho: μ ≤ $42,000 • Ha: μ > $42,000 (given claim is “more than”) • Alpha • α = 0.05 (given) • Model • Normal (n ≥ 30 and it’s a mean)

24. Example 2 (cont.) • Test statistic and p-value:

25. Example 2 (cont.) • Decision • 0.0934 > 0.05 (p-value > alpha) • DO NOT REJECT Ho • Conclusion • We do not have evidence to suggest that μ > $42,000.

26. Example 3 – Hypothesis Testing A physician claims that joggers’ maximal volume oxygen uptake is greater than the average of all adults. A sample of 15 joggers has a mean of 40.6 milliliters per kilogram (ml/kg) and a standard deviation of 6 ml/kg. If the average of all adults is 36.7 ml/kg, is there enough evidence to support the physicians claim at α = 0.05?

27. Example 3 (cont.) • Hypotheses • Ho: μ ≤ 36.7 • Ha: μ > 36.7 • Alpha • α = 0.05 (given) • Model • t(14)

28. Example 3 (cont.) • Test statistic and p-value:

29. Example 3 (cont.) • Decision • (0.01,0.025) < 0.05 (p-value < alpha) • REJECT Ho • Conclusion • There is evidence to suggest that μ > 36.7.

30. Example 4 – Hypothesis Testing A researcher knows from past studies that the standard deviation of the time it takes to inspect a car is 16.8 minutes. A sample of 24 cars is selected and inspected. The standard deviation was 12.5 minutes. At α=0.05, can it be concluded that the standard deviation has changed?

31. Example 4 (cont.) • Hypotheses • Ho: σ = 16.8 • Ha: σ≠ 16.8 • Alpha • α = 0.05 (given) • Model • χ2(23)

32. Example 4 (cont.) • Test statistic and p-value:

33. Example 4 (cont.) • Decision • (0.05,0.10) > 0.05 (p-value > alpha) • DO NOT REJECT Ho • Conclusion • There is not enough evidence to suggest that σ≠ 16.8.