Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 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

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2 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.

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 3 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

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4 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)

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 5 In a P-value method, 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 

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 6 Testing Claims with Confidence Intervals A confidence interval estimate of a population parameter contains the likely values of that parameter. We should therefore reject a claim that the population parameter has a value that is not included in the confidence interval.

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 7 Testing Claims with Confidence Intervals  95% confidence interval of 106 body temperature data (that is, 95% of samples would contain true value µ )  98.08º < µ < 98.32º  98.6º is not in this interval  Therefore it is very unlikely that µ = 98.6º  Thus we reject claim µ = 98.6 Claim: mean body temperature = 98.6°, where n = 106, x = 98.2° and s = 0.62°

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 8 Underlying Rationale of Hypotheses Testing When testing a claim, we make an assumption (null hypothesis) that contains equality. We then compare the assumption and the sample results and we form one of the following conclusions:  If the sample results can easily occur when the assumption is true, we attribute the relatively small discrepancy between the assumption and the sample results to chance.  If the sample results cannot easily occur when the assumption is true, we explain the relatively large discrepancy between the assumption and the sample by concluding that the assumption is not true.

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 9 Testing a Claim about a Mean: Small Samples Section 7-4 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics, Addison Wesley Longman

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 10 Figure 7-10 Choosing between the Normal and Student t- Distributions when Testing a Claim about a Population Mean µ Is n > 30 ? Is the distribution of the population essentially normal ? (Use a histogram.) No Yes No Start Is  known ? Use normal distribution with x – µ x  / n Z  (If  is unknown use s instead.) Use non-parametric methods, which don’t require a normal distribution. Use normal distribution with x – µ x  / n Z  (This case is rare.) Use the Student t-distribution with x – µ x s / n tt

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 11 a) For any population distribution (shape) where, n > 30 (use s for  if  is unknown) b) For a normal population distribution where 1)   is known ( n any size), or 2)   is unknown and n > 30 (use s for  ) The distribution of sample means will be NORMAL and you can use Table A-2 :

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 12 The Student t -distribution and Table A-3 should be used when:  Population distribution is essentially normal   is unknown  n  30

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 13 Test Statistic ( TS ) for a Student t -distribution Critical Values ( CV )  Found in Table A-3 Formula card, back book cover, or Appendix  Degrees of freedom (df) = n – 1  Critical t values to the left of the mean are negative t = x – µ x s n

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14 Important Properties of the Student t Distribution 1.The Student t distribution is different for different sample sizes (see Figure 6-5 in Section 6-3). 2.The Student t distribution has the same general bell shape as the normal distribution; its wider shape reflects the greater variability that is expected with small samples. 3.The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0). 4.The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a   = 1). 5.As the sample size n gets larger, the Student t distribution get closer to the normal distribution. For values of n > 30, the differences are so small that we can use the z values instead of developing a much larger table of critical t values. (The values in the bottom row of Table A-3 are equal to the corresponding critical z values from the standard normal distribution.)

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 15 All three methods 1) Traditional method 2) P-value method 3) Confidence intervals and the testing procedure Step 1 to Step 8 are still valid, except that the test statistic (therefore corresponding Table) is different.

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 16 The larger Student t -distribution values shows that with small samples the sample evidence must be more extreme before the difference is significant.

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 17 P -Value Method  Table A-3 includes only selected values of   Specific P -values usually cannot be found  Use Table to identify limits that contain the P -value  Some calculators and computer programs will find exact P -values

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 18 Example Conjecture: “the average starting salary for a computer science gradate is $30,000 per year”. For a randomly picked group of 25 computer science graduates, their average starting salary is $36,100 and the sample standard deviation is $8,000.

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 19 Example Solution Step 1: µ = 30k Step 2: µ > 30k (if believe to be no less than 30k) Step 3: H 0: µ = 30k versus H 1: µ > 30k Step 4: Select  = 0.05 (significance level) Step 5: The sample mean is relevant to this test and its sampling distribution is t-distribution with ( ) = 24 degrees of freedom.

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 20 t-distribution (DF = 24) Assume the conjecture is true! t =t = x – µxx – µx SS n Test Statistic: Critical value = 1.71 * 8000/ = K ( t = 0) Fail to reject H 0 Reject H k ( t = 1.71 ) (Step 6)

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 21 t-distribution (DF = 24) Assume the conjecture is true! t =t = x – µxx – µx ss n Test Statistic: Critical value = 1.71 * 8000/ = K ( t = 0) Fail to reject H 0 Reject H k ( t = 1.71 ) Sample data: t = x = 36.1k or (Step 7)

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 22 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:

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 23 t-distribution (DF = 24) Assume the conjecture is true! t =t = x – µxx – µx S  n Test Statistic: 30 K 36.1 k (Step 6) P-value = area to the right of the test statistic t = / 5 = P-value = (by a computer program)

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 24 t-distribution (DF = 24) 30 K 36.1 k (Step 7) P-value = area to the right of the test statistic t = / 5 = P-value = P-value < 0.01 Highly statistically significant (Very strong evidence against the null hypothesis)