2. 2 KNR 445 Statistics Hyp-tests Slide 1 Stage 5: The test statistic!  So, we insert that threshold value, and now we are asked for some more values… The sample mean The sample size The population SD 1 3

3. KNR 445 Statistics Hyp-tests Slide 2 Stage 5: The test statistic!  Why do we need these three? Because now we have to convert our difference score to a score on the distribution of sample means  Remember this? The purpose of this statistic was to convert a raw score difference (from the mean) by scaling it according to the spread of raw scores in the distribution of raw scores 1

4. KNR 445 Statistics Hyp-tests Slide 3 Stage 5: The test statistic! Sample mean Population mean Variability of sample means The purpose of this statistic is the same, but it converts a sample mean difference (from µ) by scaling it according to the spread of all sample means in the distribution of sample means 1 2 3

5. KNR 445 Statistics Hyp-tests Slide 4 Stage 5: The test statistic!  Understanding influences on the distribution of sample means…we’ll use the applet again Note sample size… & note spread of sample means 1

6. KNR 445 Statistics Hyp-tests Slide 5 Stage 5: The test statistic!  Understanding influences on the distribution of sample means…we’ll use the applet again As sample size goes up… Spread of sample means goes down 1

7. KNR 445 Statistics Hyp-tests Slide 6 Stage 5: The test statistic!  Understanding influences on the distribution of sample means…  That means that the test statistic has to take sample size into account  Other influences are mean difference (sample – population) and variability in the population  How do you think each of these things influence the test statistic?  This will help you understand why the test statistic looks like it does 1 2 3 4

8. KNR 445 Statistics Hyp-tests Slide 7 Stage 5: The test statistic! Sample mean Population mean Variability of sample means A closer look: to understand how the mean difference, population variance, and sample size affect the test statistic, we need to look at the SE M in more detail 1

9. KNR 445 Statistics Hyp-tests Slide 8 Stage 5: The test statistic! Population standard deviation Sample size So…can you see the influences? 1 23 4

10. KNR 445 Statistics Hyp-tests Slide 9 Stage 5: The test statistic!  To calculate, then…  First the standard error of the mean:  Now the test statistic itself: 1

11. KNR 445 Statistics Hyp-tests Slide 10 Stage 5: The test statistic!  For you to practice, I’ve provided a simple excel file that does the calculation bit for you… 1

12. KNR 445 Statistics Hyp-tests Slide 11 Stage 6: The comparison & decision  Do we fail to reject the null? Or reject the null? 1

13. KNR 445 Statistics Hyp-tests Slide 12 3 ways of phrasing the decision…  What is the probability of obtaining a Z obs = 1.273 if the difference is attributable only to random sampling error?  Is the observed probability (p) less than or equal to the  -level set?  Is p   ? 1

14. KNR 445 Statistics Hyp-tests Slide 13 Reporting the Results  The observed mean of our treatment group was 51.88 (  13.62) pages per employee per week. The z-test for one sample indicates that the difference between the observed mean of 51.88 and the population average of 49.52 was not statistically significant (Z obs = 1.27, p > 0.1). Our sample of employees did not use significantly more paper than the norm. 1

15. KNR 445 Statistics Hyp-tests Slide 14 Do not reject H 0 vs. Accept H 0  Accept infers that we are sure Ho is valid  Do not reject implies that this time we are unable to say with a high enough degree of confidence that the difference observed is attributable to anything other than sampling error. 1 2