Chapter 8 Inferences Based on a Single Sample: Tests of Hypothesis

2 The Elements of a Test of Hypothesis 7 elements 1.The Null hypothesis 2.The alternate, or research hypothesis 3.The test statistic 4.The rejection region 5.The assumptions 6.The Experiment and test statistic calculation 7.The Conclusion

3 The Elements of a Test of Hypothesis Does a manufacturer’s pipe meet building code? Null hypothesis – Pipe does not meet code (H 0 ):  < 2400 Alternate hypothesis – Pipe meets specifications (H a ):  > 2400

4 The Elements of a Test of Hypothesis Test statistic to be used Rejection region Determined by Type I error, which is the probability of rejecting the null hypothesis when it is true, which is . Here, we set  =.05 Region is z>1.645, from z value table

5 The Elements of a Test of Hypothesis Assume that s is a good approximation of  Sample of 60 taken,, s=200 Test statistic is Test statistic lies in rejection region, therefore we reject H 0 and accept H a that the pipe meets building code

6 The Elements of a Test of Hypothesis Type I vs Type II Error Conclusions and Consequences for a Test of Hypothesis True State of Nature ConclusionH 0 TrueH a True Accept H 0 (Assume H 0 True) Correct decisionType II error (probability  ) Reject H 0 (Assume H a True) Type I error (probability  ) Correct decision

7 The Elements of a Test of Hypothesis 1.The Null hypothesis – the status quo. What we will accept unless proven otherwise. Stated as H 0 : parameter = value 2.The Alternative (research) hypothesis (H a ) – theory that contradicts H 0. Will be accepted if there is evidence to establish its truth 3.Test Statistic – sample statistic used to determine whether or not to reject Ho and accept H a

8 The Elements of a Test of Hypothesis 4.The rejection region – the region that will lead to H 0 being rejected and H a accepted. Set to minimize the likelihood of a Type I error 5.The assumptions – clear statements about the population being sampled 6.The Experiment and test statistic calculation – performance of sampling and calculation of value of test statistic 7.The Conclusion – decision to (not) reject H 0, based on a comparison of test statistic to rejection region

9 Large-Sample Test of Hypothesis about a Population Mean Null hypothesis is the status quo, expressed in one of three forms H 0 :  = 2400 H 0 :  ≤ 2400 H 0 :  ≥ 2400 It represents what must be accepted if the alternative hypothesis is not accepted as a result of the hypothesis test

10 Large-Sample Test of Hypothesis about a Population Mean Alternative hypothesis can take one of 3 forms: One-tailed, upper tail H a :  <2400 One-tailed, upper tail H a :  >2400 Two-tailed H a :  2400

11 Large-Sample Test of Hypothesis about a Population Mean

12 Large-Sample Test of Hypothesis about a Population Mean If we have: n=100, = 11.85, s =.5, and we want to test if  12 with a 99% confidence level, our setup would be as follows: H 0 : = 12 H a :  12 Test statistic Rejection region z (two-tailed)

13 Large-Sample Test of Hypothesis about a Population Mean CLT applies, therefore no assumptions about population are needed Solve Since z falls in the rejection region, we conclude that at.01 level of significance the observed mean differs significantly from 12

14 Observed Significance Levels: p- Values The p-value, or observed significance level, is the smallest  that can be set that will result in the research hypothesis being accepted.

15 Observed Significance Levels: p- Values Steps: Determine value of test statistic z The p-value is the area to the right of z if H a is one-tailed, upper tailed The p-value is the area to the left of z if H a is one-tailed, lower tailed The p-valued is twice the tail area beyond z if H a is two-tailed.

16 Observed Significance Levels: p- Values When p-values are used, results are reported by setting the maximum  you are willing to tolerate, and comparing p-value to that to reject or not reject H 0

17 Small-Sample Test of Hypothesis about a Population Mean When sample size is small (<30) we use a different sampling distribution for determining the rejection region and we calculate a different test statistic The t-statistic and t distribution are used in cases of a small sample test of hypothesis about  All steps of the test are the same, and an assumption about the population distribution is now necessary, since CLT does not apply

18 Small-Sample Test of Hypothesis about a Population Mean where t  and t  /2 are based on (n-1) degrees of freedom Rejection region: (or when H a : Test Statistic: Ha:Ha:H a : (or H a: ) H0:H0:H0:H0: Two-Tailed TestOne-Tailed Test Small-Sample Test of Hypothesis about

19 Large-Sample Test of Hypothesis about a Population Proportion Rejection region: (or when where, according to H 0, and Test Statistic: Ha:Ha:H a : (or H a: ) H0:H0:H0:H0: Two-Tailed TestOne-Tailed Test Large-Sample Test of Hypothesis about

20 Large-Sample Test of Hypothesis about a Population Proportion Assumptions needed for a Valid Large-Sample Test of Hypothesis for p A random sample is selected from a binomial population The sample size n is large (condition satisfied if falls between 0 and 1

21 Calculating Type II Error Probabilities: More about  Type II error is associated with , which is the probability that we will accept H 0 when H a is true Calculating a value for  can only be done if we assume a true value for  There is a different value of  for every value of 

22 Calculating Type II Error Probabilities: More about  Steps for calculating  for a Large-Sample Test about  1.Calculate the value(s) of corresponding to the borders of the rejection region using one of the following: Upper-tailed test: Lower-tailed test: Two-tailed test:

23 Calculating Type II Error Probabilities: More about  2.Specify the value of in H a for which  is to be calculated. 3.Convert border values of to z values using the mean, and the formula 4.Sketch the alternate distribution, shade the area in the acceptance region and use the z statistics and table to find the shaded area, 

24 Calculating Type II Error Probabilities: More about  The Power of a test – the probability that the test will correctly lead to the rejection of H 0 for a particular value of  in H a. Power is calculated as 1- .

25 Tests of Hypothesis about a Population Variance Hypotheses about the variance use the Chi- Square distribution and statistic The quantity has a sampling distribution that follows the chi-square distribution assuming the population the sample is drawn from is normally distributed.

26 Tests of Hypothesis about a Population Variance where is the hypothesized variance and the distribution of is based on (n-1) degrees of freedom Rejection region: Or Rejection region: (or when H a : Test Statistic: Ha:Ha:H a : (or H a: ) H0:H0:H0:H0: Two-Tailed TestOne-Tailed Test Small-Sample Test of Hypothesis about