Hypothesis testing Chapter S12 Learning Objectives Understand the principles of statistical inference Formulate null and alternative hypotheses Understand one-tailed and two-tailed tests Understand type I and type II errors Understand test statistics Understand the significance level of a test Understand and calculate critical values Understand the regions of acceptance and rejection © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

Hypothesis testing Chapter S12 Learning Objectives continued ... Calculate and interpret a one-sample z-test statistic Calculate and interpret a one-sample t-test statistic Calculate and interpret a paired t-test statistic Calculate and interpret a two-sample t-test statistic Understand and calculate p-values © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

Statistical inference One of the major roles of statisticians is to draw conclusions from data This is referred to as statistical inference We can put a probability on whether a conclusion is correct within reasonable doubt Conclusions can always be wrong Statistical inference plays a major role in decision making © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

Statistical inference Decision-making process: 1. Collect the data 2. Summarise the data (using either visual displays or descriptive statistics) 3.Set up an hypothesis (i.e. claim or theory) to be tested 4.Calculate the probability of obtaining a sample such as the one we have if the hypothesis is true 5. Either accept or reject the hypothesis Conclusions will be made based on samples taken from the population © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

Ho:(statement) v H1(alternative statement) The null hypothesis (questions dealing with differences between samples) Technique for dealing with these problems begins with the formulation of an hypothesis. Null hypothesis is a statement that nothing unusual has occurred. The notation is Ho. Alternative hypothesis states that something unusual has occurred. The notation is H1 or HA Together they may be written in the form: Ho:(statement) v H1(alternative statement) Where: ‘v’ stands for versus © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

Alternative hypothesis May be classified as two-tailed test or one-tailed test Two-tailed test (two sided alternative) Test with no preconceived notion that the true value of µ is either above or below the hypothesised value of µ H1: µ  µo © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

Errors There are two possible errors in making a conclusion about a null hypothesis 1. Type I errors occur when you reject Ho as being false when Ho is really true 2. Type II errors occur when you accept Ho as being true when Ho is really false © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

Significance level This level represents the borderline probability between whether an event has occurred by chance or whether an unusual event has taken place Most common significance level used is 0.05, commonly written as = 0.05 5% significance level says in effect that an event that occurs less than 5% of the time is considered unusual © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

Test statistics one sample z-test one sample t-test Test for determining if a single sample from a population is consistent with the rest of the population one sample z-test one sample t-test © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

Value of a test statistic Information required in calculation: 1. the size (n) of the sample 2. the mean ( ) of the sample 3. the standard deviation (s) of the sample Other information of interest might include: 1. Does the population have a normal distribution? 2. Is the population’s standard deviation known? 3. Is the sample size (n) large? © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

Drawing a conclusion The steps that should be undertaken to perform a one-sample test are: 1. Set up null and alternative hypotheses—including deciding whether to use a one-sided or two-sided test 2. Decide on significance level you are using 3. Write down the relevant data 4. Decide on the test statistic to be used 5. Calculate the value of the test statistic 6. Find the relevant critical value and decide whether H0 is to be not rejected or rejected 7. Draw an appropriate conclusion © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

Test statistics (two sample problems) Paired t-test two samples that are to be compared with each other often referred to as two-sample problems have a structure such that the data are paired samples must each contain the same number of observations © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

Test statistics (two sample problems) Two-sample t-test Samples are not paired, they are independent Two samples need not contain the same number of observations Most common test statistic used in this situation is a two-sample t-test Also known as a pooled test Calculation requires more work than for the paired t-test © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

Test statistics (two sample problems) p-values Alternative to using critical values for testing hypotheses Calculates the probability of obtaining a value as extreme as the value of the test statistic © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher