1. Lesson 11 - R Review of Testing a Claim

2. Objectives Explain the logic of significance testing. List and explain the differences between a null hypothesis and an alternative hypothesis. Discuss the meaning of statistical significance. Use the Inference Toolbox to conduct a large sample test for a population mean. Compare two-sided significance tests and confidence intervals when doing inference. Differentiate between statistical and practical “significance.” Explain, and distinguish between, two types of errors in hypothesis testing. Define and discuss the power of a test.

3. Vocabulary none new

4. What you Learned –State the null and alternative hypotheses in a testing situation when the parameter in question is a population mean µ. –Explain in nontechnical language the meaning of the P-value when you are given the numerical value of P for a test. –Calculate the one-sample z-statistic and the P-value for both one-sided and two-sided tests about the mean µ of a Normal population. –Assess statistical significance at standard levels α by comparing P to α. –Recognize that significance testing does not measure the size or importance of an effect. –Recognize when you can use the z test and when the data collection design or a small sample from a skewed population makes it inappropriate. –Explain Type I error, Type II error, and power in a significance- testing problem.

5. Hypothesis Testing Approaches Classical –Logic: If the sample mean is too many standard deviations from the mean stated in the null hypothesis, then we reject the null hypothesis (accept the alternative) P-Value –Logic: Assuming H 0 is true, if the probability of getting a sample mean as extreme or more extreme than the one obtained is small, then we reject the null hypothesis (accept the alternative). Confidence Intervals –Logic: If the sample mean lies in the confidence interval about the status quo, then we fail to reject the null hypothesis

6. Determining H o and H a H o – is the status quo; what the situation is currently the claim made by the manufacturer H a – is the alternative that you are testing; the new idea the thing that proves the claim false H 0 and H a always refer to population parameters

7. Inference Toolbox Step 1: Hypothesis –Identify population of interest and parameter –State H 0 and H a Step 2: Conditions –Check appropriate conditions Step 3: Calculations –State test or test statistic –Use calculator to calculate test statistic and p-value Step 4: Interpretation –Interpret the p-value (fail-to-reject or reject) –Don’t forget 3 C’s: conclusion, connection and context

8. Conditions for Significance Tests SRS –simple random sample from population of interest Normality –For means: population normal or large enough sample size for CLT to apply or use t-procedures –t-procedures: boxplot or normality plot to check for shape and any outliers (outliers is a killer) –For proportions: np ≥ 10 and n(1-p) ≥ 10 Independence –Population, N, such that N > 10n

9. Using Your Calculator: Z-Test For classical or p-value approaches Press STAT –Tab over to TESTS –Select Z-Test and ENTER Highlight Stats Entry μ 0, σ, x-bar, and n from summary stats Highlight test type (two-sided, left, or right) Highlight Calculate and ENTER Read z-critical and/or p-value off screen X-bar – μ Z 0 = --------------- σ / √n

10. Reality H 0 is TrueH 1 is True Conclusion Do Not Reject H 0 Correct Conclusion Type II Error Reject H 0 Type I Error Correct Conclusion H 0 : the defendant is innocent H 1 : the defendant is guilty Type I Error (α): convict an innocent person Type II Error (β): let a guilty person go free Note: a defendant is never declared innocent; just not guilty decrease α  increase β increase α  decrease β Hypothesis Testing: Four Outcomes

11. Increasing the Power of a Test Four Main Methods: –Increase significance level,  –Consider a particular alternative that is farther away from  –Increase the sample size, n, in the experiment –Decrease the population (or sample) standard deviation, σ Only increasing the sample size and the significance level are under the control of the researcher

12. Summary and Homework Summary –Inference testing has an H 0 and an H a to evaluate –H 0 is the status quo and cannot be proven –H a is the claim that’s being tested –Small p-values (tails), or p < , are in favor of H a –P(Type I error – reject H 0 when H 0 is true) =  –P(Type II error – FTR H 0 when H a is true) =  –Power (of a test) = 1 -  –Increase power by increasing n or  Homework –pg 738 – 9; 11.65 – 74

13. Problem 1 When performing inference procedures for means we sometimes use the normal distribution and sometimes use Student’s t distribution. How do you decide which to use? We use normal distributions in inference procedures for population proportions always and for population means if we know the population standard deviation (very rare). We use the Student’s t distribution for population means when we don’t know the population standard deviation and have to use the sample standard deviation as its estimate.

14. Problem 2a The blood sugar levels of 7 randomly selected rabbits of a certain species are measured 90 minutes after injection of 0.8 units of insulin. The data (in mg/100 ml) are: 45, 34, 32, 48, 39, 45, 51. The researcher would like to use this sample to make inferences about the mean blood sugar level for the population of rabbits. (a) List the conditions (assumptions) that must be satisfied in order to compute confidence intervals or perform hypothesis tests. Also, provide information to convince me that the condition is satisfied. Conditions: SRS – problem states random selection, so assume ok Normality – n = 7 is too small for CLT to apply so box-plot of data is required to check if normality is not a good assumption and to see if there are any outliers! [no outliers, symmetry not great, normality plot ok] Independence – assumption is reasonable from problem

15. Problem 2b The blood sugar levels of 7 randomly selected rabbits of a certain species are measured 90 minutes after injection of 0.8 units of insulin. The data (in mg/100 ml) are: 45, 34, 32, 48, 39, 45, 51. The researcher would like to use this sample to make inferences about the mean blood sugar level for the population of rabbits. (b) Calculate a 90% confidence interval for the mean blood sugar level in the population of all rabbits of this species 90 minutes after insulin injection. Calculations: n = 7, x-bar = 42 mg/100 ml, s = 7.165 mg/100 ml, and α/2 =.05 CI = x-bar  t.95,6 (s/√n) = 42  1.943(7.165/√7) = 42  5.262 From calculator: [36.738, 47.262 ]

16. Problem 3 The mean length of the incision used in arthroscopic knee surgery has been 2.0 cm. An article in a recent medical journal reports that a new procedure can reduce the average length of the incision used in such surgeries. The authors of the article reported that in a sample of 36 such surgeries in which the new technique was used, the average incision length was 1.96 cm, with a standard deviation of 0.13 cm. Perform a test at the α =.05 level to determine whether the mean length of incision used in the new procedure is significantly less than 2.0 cm. Assume that any conditions needed for inference are satisfied.

17. Problem 3 Hypothesis: μ = average incision length of old procedure H 0 : μ = 2.0 cm no change H a : μ < 2.0 cm new procedure better (One sided test) Conditions: “Assume that any conditions... satisfied” Calculations: μ = 2.0 cm, n = 36, x-bar = 1.96 cm, s = 0.13 cm, and α =.05 TS = (1.96 – 2.0)/(0.13/√36) = -1.846 From calculator: TS = -1.846 p-value = 0.0367 Interpretation: Since p < , then we have evidence to reject H 0 in favor of H a. The new procedure has smaller average length incision (scars) than the old procedure. Statistically significant result, but not much real practical significance (0.04 cm improvement or about 1/50 of an inch).

18. Problem 4 A company institutes an exercise break for its workers to see if this will improve job satisfaction. A questionnaire designed to assess worker satisfaction was given to 10 randomly selected workers before and after implementation of the exercise program. These scores are provided in the table below: Perform a significance test to determine whether the exercise program was effective in improving job satisfaction. You should state the conditions necessary for the test you perform to be valid, but you do not need to check these conditions. Organize your work clearly and be sure to show all steps of the test below. Worker #12345678910 Score before3428294526272415 27 Score after33365041374139212037

19. Problem 4 cont Worker #12345678910 Score before3428294526272415 27 Score after33365041374139212037 Difference821-41114156510 Hypothesis: μ = average difference between after and before program H 0 : μ diff = 0 program ineffective H a : μ diff > 0 program improves job sat. (One sided test) Conditions: SRS – ok; Normal – no CLT, but box-plot and normality plots OK; Independent – weak Calculations: μ = 0, n = 10, x-bar = 8.94, s = 6.749, and α =.05 TS = (8.94 – 0)/(6.749/√10) = 4.189 From calculator: t = 4.189 and p-value = 0.0012 Interpretation: Since p-value is < , we reject H 0 in favor of H a and conclude that the exercise program improves job satisfaction.