1. BPS - 3rd Ed. Chapter 141 Tests of significance: the basics

2. BPS - 3rd Ed. Chapter 142 u What would happen if we repeated the sample or experiment many times? u How likely would it be to see the observed results if the claim was untrue? u Do the data give evidence against a claim? Reasoning of Tests of Significance

3. BPS - 3rd Ed. Chapter 143 u The statement being tested is called the null hypothesis u The null hypothesis is a statement of “no effect” or “no difference” u The test is designed to assess the strength of evidence against the null hypothesis. Stating Hypotheses Null Hypothesis, H 0

4. BPS - 3rd Ed. Chapter 144 u The statement we are trying to find evidence for is called the alternative hypothesis. u The alternative hypothesis usually indicates “an effect” or “difference” u The alternative hypothesis expresses the hopes or suspicions we bring to the data. Stating Hypotheses Alternative Hypothesis, H a

5. BPS - 3rd Ed. Chapter 145 Case Study I: “Weight Gain” Suppose we know that weight gain after the age of 30 varies from individual to individual according to a Normal distribution with standard deviation  = 1 lbs The symbol  represents the mean or expected weight gain for all individuals (the parameter) Ten individuals between the age of 30 and 40 yield an average gains of 1.02 lbs. Do these data provide sufficient evidence that people in this age range tend to gain weight each year?

6. BPS - 3rd Ed. Chapter 146 Case Study I: “Weight Gain”  If the claim that  = 0 is true (no average weight gain), the sampling distribution of from 10 individuals is Normal with  = 0 and standard deviation  The data yielded = 1.02, which is more than three standard deviations from  = 0. This provides strong evidence that people gain weight.  If the data yielded = 0.3, which is less than one standard deviations from  = 0, there would be less convincing evidence that individuals gains weight.

7. BPS - 3rd Ed. Chapter 147 Case Study I Weight gain

8. BPS - 3rd Ed. Chapter 148 Statistical hypotheses  Null: H 0 :  =  0 u One sided alternatives H a :   H a :   u Two sided alternative H a :  0

9. BPS - 3rd Ed. Chapter 149 Weight gain Case Study I The null hypothesis is “no average weight gain” The alternative hypothesis “yes, average weight gain” H 0 :  = 0H a :  > 0 We use a one-sided test because we are interested only in determining weight gain (and not weight loss)

10. BPS - 3rd Ed. Chapter 1410 Take an SRS of size n from a Normal population with unknown mean  and known standard deviation . The test statistic for null hypothesis H 0 :  =  0 is Test Statistic Testing the Mean of a Normal Population

11. BPS - 3rd Ed. Chapter 1411 Weight Gain Case Study I The test statistic for no average weight gain is: This shows that the sample mean is more than 3 standard deviations above the hypothesized mean of 0. This provides strong evidence against H 0.

12. BPS - 3rd Ed. Chapter 1412 The P-value provides the probability that the test statistic would take a value as extreme or more extreme than the value observed if the null hypothesis were true. The smaller the P-value, the stronger the evidence the data provide against the null hypothesis P-value

13. BPS - 3rd Ed. Chapter 1413 P-value for Testing Means  H a :  >  0 v P-value is the probability of getting a value as large or larger than the observed test statistic (z) value.  H a :  <  0 v P-value is the probability of getting a value as small or smaller than the observed test statistic (z) value.  H a :  0 v P-value is two times the probability of getting a value as large or larger than the absolute value of the observed test statistic (z) value.

14. BPS - 3rd Ed. Chapter 1414 Weight Gain Case Study I For test statistic z = 3.23 and alternative hypothesis H a :  > 0, the P-value is: P-value = P(Z > 3.23) = 1 – 0.9994 = 0.0006 Interpretations: If H 0 is true, there is only a 0.0006 (0.06%) chance that we would see results at least as extreme as those in the sample  we therefore have evidence against H 0 and in favor of H a.

15. BPS - 3rd Ed. Chapter 1415 Weight gain Case Study I

16. BPS - 3rd Ed. Chapter 1416  If the P-value is as small or smaller than the significance level  (i.e., P-value ≤  ), then we say that data are statistically significant at level .  If we choose  = 0.05, we are requiring that the data give evidence against H 0 so strong that it would occur no more than 5% of the time when H 0 is true.  If we choose  = 0.01, we are insisting on stronger evidence against H 0, evidence so strong that it would occur only 1% of the time when H 0 is true. Statistical Significance

17. BPS - 3rd Ed. Chapter 1417 The four steps in carrying out a significance test: 1. State the null and alternative hypotheses. 2. Calculate the test statistic. 3. Find the P-value. 4. State your conclusion. The procedure for Steps 2 and 3 is on the next page. Tests for a Population Mean

18. BPS - 3rd Ed. Chapter 1418

19. BPS - 3rd Ed. Chapter 1419 Weight gain problem Case Study I 1. Hypotheses:H 0 :  = 0 H a :  > 0 2. Test Statistic: 3. P-value:P-value = P(Z > 3.23) = 1 – 0.9994 = 0.0006 4. Conclusion: Since the P-value is smaller than  = 0.01, there is strong evidence that people gain weight in this age range.

20. BPS - 3rd Ed. Chapter 1420 Studying Job Satisfaction Case Study II Does the job satisfaction of assembly workers differ when their work is machine-paced rather than self-paced? A matched pairs study was performed on a sample of workers, and each worker’s satisfaction was assessed after working in each setting. The response variable is the difference in satisfaction scores, self- paced minus machine-paced.

21. BPS - 3rd Ed. Chapter 1421 Studying Job Satisfaction Case Study II The null hypothesis “no average difference” in the population of assembly workers. The alternative hypothesis (that which we want to show is likely to be true) is “there is an average difference in scores” in the population of assembly workers. H 0 :  = 0H a :  ≠ 0 This is considered a two-sided test because we are interested determining a difference in either direction.

22. BPS - 3rd Ed. Chapter 1422 Studying Job Satisfaction Case Study II Suppose job satisfaction scores follow a Normal distribution with standard deviation  = 60. Data from 18 workers gave a sample mean score of 17. If the null hypothesis is µ 0 = 0, the test statistic is:

23. BPS - 3rd Ed. Chapter 1423 Studying Job Satisfaction Case Study II For test statistic z = 1.20 and alternative hypothesis H a :  ≠ 0, the P-value would be: P-value= P(Z 1.20) = 2 P(Z 1.20) = (2)(0.1151) = 0.2302 If H 0 is true, there is a 0.2302 (23.02%) chance that we would see results at least as extreme as those in the sample. Therefore do not have good evidence against H 0 and in favor of H a.

24. BPS - 3rd Ed. Chapter 1424 Studying Job Satisfaction Case Study II

25. BPS - 3rd Ed. Chapter 1425 Studying Job Satisfaction Case Study II 1. Hypotheses:H 0 :  = 0 H a :  ≠ 0 2. Test Statistic: 3. P-value:P-value = 2P(Z > 1.20) = (2)(1 – 0.8849) = 0.2302 4. Conclusion: Since the P-value is larger than  = 0.10, there is not sufficient evidence that mean job satisfaction of assembly workers differs when their work is machine-paced rather than self-paced.

26. BPS - 3rd Ed. Chapter 1426 Confidence Intervals & Two-Sided Tests A level  two-sided significance test rejects the null hypothesis H 0 :  =  0 exactly when the value  0 falls outside a level 1 –  confidence interval for .

27. BPS - 3rd Ed. Chapter 1427 Case Study II A 90% confidence interval for  is: Since  0 = 0 is in this confidence interval, it is plausible that the true value of  is 0. Thus, there is insufficient evidence (at  = 0.10) that the mean job satisfaction of assembly workers differs when their work is machine-paced rather than self-paced. Studying Job Satisfaction