1. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 1 Testing a Claim about a Mean: Large Samples Section 7-3 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman

2. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 2 Assumptions 1) Sample is large (n > 30) a) Central limit theorem applies b) Can use normal distribution 2)Can use sample standard deviation s as estimate for  if  is unknown For testing a claim about the mean of a single population

3. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 3 Three Methods Discussed 1) Traditional method 2) P-value method 3) Confidence intervals Note: These three methods are equivalent, I.e., they will provide the same conclusions.

4. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 4 Traditional (or Classical) Method of Testing Hypotheses Goal Identify: whether a sample result that is significantly different from the claimed value

5. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 5 Procedure 1. Identify the specific claim or hypothesis to be tested, and put it in symbolic form. 2. Give the symbolic form that must be true when the original claim is false. 3. Of the two symbolic expressions obtained so far, put the one you plan to reject in the null hypothesis H 0 (make the formula with equality). H 1 is the other statement. Or, One simplified rule suggested in the textbook: let null hypothesis H 0 be the one that contains the condition of equality. H 1 is the other statement. Figure 7-4

6. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 6 4. Select the significant level  based on the seriousness of a type I error. Make  small if the consequences of rejecting a true H 0 are severe. The values of 0.05 and 0.01 are very common. 5. Identify the statistic that is relevant to this test and its sampling distribution. 6. Determine the test statistic, the critical values, and the critical region. Draw a graph and include the test statistic, critical value(s), and critical region. 7. Reject H 0 if the test statistic is in the critical region. Fail to reject H 0 if the test statistic is not in the critical region. 8. Restate this previous decision in simple non-technical terms. (See Figure 7-2)

7. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 7 Original claim is H 0 FIGURE 7-2 Wording of Conclusions in Hypothesis Tests Do you reject H 0 ?. Yes (Reject H 0 ) “There is sufficient evidence to warrant rejection of the claim that... (original claim).” “There is not sufficient evidence to warrant rejection of the claim that... (original claim).” “The sample data supports the claim that... (original claim).” “There is not sufficient evidence to support the claim that... (original claim).” Do you reject H 0 ? Yes (Reject H 0 ) No (Fail to reject H 0 ) No (Fail to reject H 0 ) (This is the only case in which the original claim is rejected). (This is the only case in which the original claim is supported). Original claim is H 1

8. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 8 The traditional (or classical) method of hypothesis testing is actually comparing the sample test statistic value with the critical region value.

9. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 9 Decision Criterion (Step 7) Reject the null hypothesis if the test statistic is in the critical region Fail to reject the null hypothesis if the test statistic is not in the critical region

10. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 10 Fail to reject H 0 Reject H 0 The traditional (or classical) method of hypothesis testing is actually comparing the sample test statistic value with the critical region value. Fail to reject H 0 Reject H 0 z CV REJECT H 0 z CV z TS

11. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 11 Fail to reject H 0 Reject H 0 The traditional (or classical) method of hypothesis testing is actually comparing the sample test statistic value with the critical region value. Fail to reject H 0 Reject H 0 Fail to reject H 0 Reject H 0 Fail to reject H 0 Reject H 0 FAIL TO REJECT H 0 z TS z CV REJECT H 0 z CV z TS

12. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 12 Test Statistic for Claims about µ when n > 30 z =z =  n x – µ x (Step 6)

13. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 13 Example Conjecture: “the average starting salary for a computer science gradate is $30,000 per year”. For a randomly picked group of 36 computer science graduates, their average starting salary is $36,100 and the sample standard deviation is $8,000.

14. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 14 Example Solution Step 1: µ = 30k Step 2: µ > 30k (if believe to be no less than 30k) Step 4: Select  = 0.05 (significance level) Step 5: The sample mean is relevant to this test and its sampling distribution is approximately normal (n = 36 large, by CLT) Step 3: H 0: µ = 30k versus H 1: µ > 30k

15. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 15 Central Limit Theorem: Assume the conjecture is true! z =z = x – µxx – µx   n Test Statistic: Critical value = 1.64 * 8000/6 + 30000 = 32186.67 30 K ( z = 0) Fail to reject H 0 Reject H 0 32.2 k ( z = 1.64 ) (Step 6)

16. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 16 Central Limit Theorem: Assume the conjecture is true! z =z = x – µxx – µx   n Test Statistic: Critical value = 1.64 * 8000/6 + 30000 = 32186.67 30 K ( z = 0) Fail to reject H 0 Reject H 0 32.2 k ( z = 1.64 ) Sample data: z = 4.575 x = 36.1k or (Step 7)

17. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 17 Example Conclusion: Based on the sample set, there is sufficient evidence to warrant rejection of the claim that “the average starting salary for a computer science gradate is $30,000 per year”. Step 8:

18. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 18 P-Value Method of Testing Hypotheses Definition  P-Value (or probability value) the probability of getting a value of the sample test statistic that is at least as extreme as the one found from the sample data, assuming that the null hypothesis is true Measures how confident we are in rejecting the null hypothesis

19. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 19 Procedure is the same except for steps 6 and 7 Step 6: Find the P-value Step 7: Report the P-value  Reject the null hypothesis if the P-value is less than or equal to the significance level   Fail to reject the null hypothesis if the P-value is greater than the significance level 

20. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 20 Highly statistically significant Very strong evidence against the null hypothesis Statistically significant Adequate evidence against the null hypothesis Insufficient evidence against the null hypothesis Less than 0.01 P - value Interpretation Greater than 0.05 0.01 to 0.05

21. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 21 Figure 7-7 Finding P-Values Is the test statistic to the right or left of center ? P-value = area to the left of the test statistic P-value = twice the area to the left of the test statistic P-value = area to the right of the test statistic Left-tailed Right-tailed Right Left Two-tailed P-value = twice the area to the right of the test statistic What type of test ? Start µ µ µ µ P-value P-value is twice this area P-value is twice this area P-value Test statistic

22. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 22 Central Limit Theorem: Assume the conjecture is true! z =z = x – µxx – µx   n Test Statistic: 30 K 36.1 k (Step 6) P-value = area to the right of the test statistic Z = 36.1 - 30 8 / 6 = 4.575 P-value =.0000024

23. Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 23 Central Limit Theorem: 30 K 36.1 k (Step 7) P-value = area to the right of the test statistic Z = 36.1 - 30 8 / 6 = 4.575 P-value =.0000024 P-value < 0.01 Highly statistically significant (Very strong evidence against the null hypothesis)