Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics S eventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Nine Part 3 (Section 9.4) Hypothesis Testing

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 2 Hypothesis Testing About a Population Mean  when Sample Evidence Comes From a Small (n < 30) Sample Use the Student’s t distribution with n – 1 degrees of freedom.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 3 Student’s t Variable Wen we draw a random sample from a population that has a mound-shaped distribution with mean , then:

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 4 C represents the level of confidence

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 5  ' is the significance level for a one-tailed test

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 6  ' is the significance level for a right-tailed test  ' = area to the right of t 0 t ''

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 7  ' is the significance level for a left-tailed test  ' = area to the left of – t – t 0 ''

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 8  '' is the significance level for a two-tailed test

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 9  '' is the significance level for a two-tailed test  ' ' = sum of the areas in the two tails – t 0 t '' ''

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 10  '' = 2  '

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 11 Find the critical value t 0 for a left-tailed test of  with n = 4 and level of significance 0.05.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 12 Find the critical value t 0 for a left-tailed test of  with n = 4 and level of significance 0.05.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 13 Find the critical value t 0 for a left-tailed test of  with n = 4 and level of significance 0.05.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 14 Find the critical value t 0 for a left-tailed test of  with n = 4 and level of significance 0.05.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 15 t = – Find the critical value t 0 for a left-tailed test of  with n = 4 and level of significance 0.05.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 16 The Critical Region for the Left-Tailed Test –  ' = 0.05

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 17 Find the critical values t 0 for a two-tailed test of  with n = 4 and level of significance 0.05.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 18 Find the critical value t 0 for a two-tailed test of  with n = 4 and level of significance 0.05.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 19 Find the critical value t 0 for a two-tailed test of  with n = 4 and level of significance 0.05.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 20 Find the critical value t 0 for a two-tailed test of  with n = 4 and level of significance 0.05.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 21 Find the critical value t 0 for a two-tailed test of  with n = 4 and level of significance t =  3.182

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 22 The Critical Region for the Two-Tailed Test  ' ' = sum of the areas in the two tails = 0.05 –  ' = 0.025

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 23 To Complete a t Test Find the critical value(s) and critical region. Convert the sample test statistic to a t value. Locate the t value on a diagram showing the critical region. If the sample t value falls in the critical region, reject H 0. If the sample t value falls outside the critical region, do not reject H 0.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 24 Use a 10% level of significance to test the claim that the mean weight of fish caught in a lake is 2.1 kg (against the alternate that the weight is lower). A sample of five fish weighed an average of 1.99 kg with a standard deviation of 0.09 kg.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 25 … test the claim that the mean weight of fish caught in a lake is 2.1 kg (against the alternate that the weight is lower).... H 0 :  = 2.1 H 1 :  < 2.1

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 26 A sample of five fish weighed... d.f. = 5 – 1 = 4

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 27 Find the critical value(s) and critical region. For a left-tailed test with  ' = 0.10 and d.f. = 4, Table 6 indicates that the critical value of t = – 1.533

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 28 The Critical Region for the Left-Tailed Test –  ' = 0.10

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 29 … A sample of five fish weighed an average of 1.99 kg with a standard deviation of 0.09 kg.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 30 When t falls within the critical region reject the null hypothesis. – 2.73 –

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 31 We conclude (at 10% level of significance) that the true weight of the fish in the lake is less than 2.1 kg.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 32 P Values for Tests of  for Small Samples The probability of getting a sample statistic as far (or farther) into the tails of the sampling distribution as the observed sample statistic. The smaller the P value, the stronger the evidence to reject H 0. Using Table 6 we find an interval containing the P value.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 33 Determine the P value when testing the claim that the mean weight of fish caught in a lake is 2.1 kg (against the alternate that the weight is lower). A sample of five fish weighed an average of 1.99 kg with a standard deviation of 0.09 kg.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 34 We completed a left-tailed test with:

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 35 When working with a left- tailed test, use  '.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 36 For t = –2.73 and d.f = 4 Sample t = 2.73

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved < P value < Sample t = 2.73

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved < P value < Since the range of P values was less than  (10%), we rejected the null hypothesis.