Stage 5: The test statistic!

1. Stage 5: The test statistic! 1 • So, we insert that threshold value, and now we are asked for some more values… The sample mean The sample size The population SD 2 3

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 scoredifference (from the mean) by scaling it according to the spread of raw scores in the distribution of raw scores 1

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

4. Stage 5: The test statistic! • Understanding influences on the distribution of sample means…we’ll use the applet again 1 Note sample size… & note spread of sample means

5. Stage 5: The test statistic! • Understanding influences on the distribution of sample means…we’ll use the applet again 1 As sample size goes up… Spread of sample means goes down

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

7. Population mean Sample mean Variability of sample means Stage 5: The test statistic! A closer look: to understand how the mean difference, population variance, and sample size affect the test statistic, we need to look at the SEM in more detail 1

8. Stage 5: The test statistic! 1 Population standard deviation So…can you see the influences? 2 3 4 Sample size

9. Stage 5: The test statistic! • To calculate, then… • First the standard error of the mean: • Now the test statistic itself: 1

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

11. Stage 6: The comparison & decision • Do we fail to reject the null? Or reject the null? 1

12. 3 ways of phrasing the decision… • What is the probability of obtaining a Zobs = 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

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 (Zobs = 1.27, p> 0.1). Our sample of employees did not use significantly more paper than the norm. 1

14. Do not reject H0 vs. Accept H0 • 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. 2 1