1. Stats 95

2. Z-Scores & Percentiles
Each z-score is associated with a percentile.
Z-scores tell us the percentile of a particular score
Can tell us % of pop. above or below a score, and the % of pop. between the score and the mean and the tail.

3. Transforming Z-score into Percentiles
DRAW A GRAPH!
Calculate z-score
Estimate the percentile of the z-score using probability distribution
Use z-score chart to transform z-score into percentile
Use graph to make sure answer makes sense
Draw a Graph!…did I mention you need to draw a graph? Yeah, draw a graph.

4. Transforming z-Scores into Percentiles
Use a chart like this in Appendix A of your text (Yes, you need the textbook) to find the percentile of you z-score.
This table gives the distance between the mean (zero) and the z-score.
To calculate cumulative percentile :
Of positive z-score (z)
Of negative z-score (-z)

6. Example: Height Jessica has a height of 66.41 inches tall (5’6’’)
The mean of the population of height for girls is 63.80
The standard deviation for the population height fir girls is 2.66
According to z-score table, the percentile associated with z = .98 is 33.65%

7. Height Example: Did I mention? DRAW A GRAPH!!
Jessica’s z-score for her height is .98, associated percentage of 33.65%.
This means
there is 33.65% of the population is between the mean and Jessica’s score.
There is a 33.65% chance of Jess being taller than the average by this amount BY CHANCE ALONE
Mean = 50th percentile, therefore to find the Jessica’s percentile = = 83.65%.
84% of the population of girls is shorter than Jessica, and there is a % = 16% chance of someone being this tall by CHANCE ALONE.

8. Hypothesis Testing
Identify Population and comparison group, state assumptions
Define the Null Hypothesis
Define the Research Hypothesis or Alternative hypothesis
Define the Research and Control Group
Define the Dependent and Independent Variables
State relevant characteristics of comparison distribution
Determine critical cutoff values
Calculate statistic
Reject or Fail to Reject the NULL Hypothesis

9. Statistical Significance
A finding is statistically significant if the data differ from what we would expect from chance alone, if there were, in fact, no actual difference.
They may not be significant in the sense of big, important differences, but they occurred with a probability below the critical cutoff value, usually a z-score or p < .05

10. Assumptions for Parametric Tests
Dependent variable is a scale variable  interval or ratio
If the dependent variable is ordinal or nominal, it is a non-parametric test
Participants are randomly selected
If there is no randomization, it is a non-parametric test (and likely a flawed experiment, or one limited in generalization)
The shape of the population of interest is approximately normal
If the shape is not normal, it is a non-parametric test

11. Assumptions for Parametric Tests

12. Hypothesis Testing with z-Scores: Example
Ursuline College, 97 students took Major Field Test in Psychology (MFTP). Is the score of this group of students statistically different* from the total population of students who took the test?
*is the probability of the difference greater than what we would expect to happen by chance alone. (i.e., how big is their horseshoe?)

14. Hypothesis Testing with z-Scores Step 1 Populations, Distribution, Assumptions
Identify Population, distributions and assumptions
All students at Ursuline who took the exam
All students nationally who took exam
Distribution
Comparison sample is not an individual but a group mean for all the students at Ursuline who took exam therefore comparison distribution is a sample mean to a population.
Assumptions
Scale is continuous interval (test scores)
Random selection unknown, so we are limited in generalizing
Shape should be normal, sample size of 97 Ursuline students, substantially larger than recommended 30

15. Hypothesis Testing with z-Scores: Step 2: State Null and research Hypothesis
Non-directional
Population Parameters
The Null Hypothesis: Ursuline Students are no different from rest of the nation.
Research Hypothesis, U. students are Different (better or worse) than population
Two-tailed test
The Null Hypothesis: Ursuline Students are not better from rest of the nation.
Research Hypothesis, U. students are better than population
One-tailed Test (Rare!)

16. Hypothesis Testing with z-Scores: Step 3: Determine Characteristics (Parameters) of the Distributions
Comparing Sample Mean to Population
Null H0 and Research H1
Because we have a sample mean, we must use the Standard Error instead of the Standard Deviation of the Population
We have Standard Deviation of Population, from which we calculate the Standard Error of Sample Means
The Mean of the distribution of Means must be equal to Pop Mean.
i.e., the average of the average of scores of all classes will be the same as the average of all students.
But the average of this class might be different from the average of all classes and the average of all students
The average of Ursuline class was The average of all other College mean samples from other college classes is
We are saying Null H. assumes the mean of this class is no different from the combined mean of all classes.

17. Hypothesis Testing with z-Scores Step 4: Determine Critical Values (Set how big a Horseshoe You Will Willingly Accept)
Convention is to set cutoffs at probability less than 5%
p < .05
Two-tailed Test, means it is 2.5% at both Tails
If it was a One-tailed Test, 5% at one Tail
Draw a Graph
We know 50% of the curve falls between the mean and each end. We know that 2.5% falls between the critical z statistic and the rest of the tail. Subtract 50%-2.5 = 47.5% . Look in chart, 47.5% is associated with the z statistics of 1.96

18. Hypothesis Testing with z-Scores: Step 5: Calculate Z Statistic
Draw a Graph!

19. Hypothesis Testing with z-Scores: Step 6: DECIDE Already!
Decision is either to Reject the Null Hypothesis or to Fail to Reject the Null Hypothesis
With a Z statistic of -.26, there was a approx. 40% of getting this score by chance alone, p =
Compare to the cutoffs
z statistics: is not more extreme than -1.96
p: >.05
We fail to reject the Null H.
The probability of Ursuline producing the score of is greater than .05
p >.05
The Ursaline mean was not smaller, than we would expect from chance alone, than the population mean.

21. What problems did Fisher have to tackle in the designing the experiment that could determine if the Lady could distinguish between milk-then-tea and tea-then-milk cups of tea? She could guess randomly with a 50% probability of being correct. She could also make an error is spite of being able to distinguish, getting 9 our of 10. How many cups to present, in what order, how much to instruct the lady, and determine beforehand the likelihood of possible outcomes.

22.  What was the nature of research like before experimental methods became established? They were idiosyncratic to each scientist, lesser scientists would produce vast amounts of data would be accumulated but would not advance knowledge, such as in the inconclusive attempts to measure the speed of light. They described their conclusion and selectively presented data that demonstrated or was indicative of the point they wanted to make, without clear procedures to replicate and not with full transparency.

23. In examining the fertility index and rainfall records at Rothamsted, what two main errors did he conclude about the indexes and how did he express results about the rainfall? He examined the competing indexes, and showed when reduced to their elemental algebra, they were all versions of the same formula. And he found that the amount of rainfall was greater than the type of fertilizer used, that the effects for fertilizer and rainfall had been confounded.

24. What did Fisher conclude scientists needed to start with to do an experiment, and what would a useful experiment do? Scientists need to start with a mathematical model of the outcome of the potential experiment, which is a set of equations in which some of the symbols stand for numbers that will be collected as data, and other symbols stand for the overall outcomes of the experiment. Then a useful experiment allows for estimation of those outcomes.