**Statistics Discussion**

Copyright © October 2004 Ted Nissen

**TABLE OF
CONTENTS**

2 Review and More Introduction

3 Central Values and the Organization of Data

8 Theoretical Distributions
Including the Normal Distribution

9 Samples and Sampling Distributions

11 The t Distribution and the
t-Test

12 Analysis of Variance: One-Way
Classification

13 Analysis of Variance: Factorial
Design

14 The Chi Square Distribution

16 Vista Formulas and Analysis

** **

9.6.6.4.2.1.1
The Web
reference below 2^{nd} applet gives you the z score to be used in the
equation above. Pug in a mean of 0 and SD (Standard Deviation) of 1, put in the
percentage in decimals (eg .10=10%, .20=20%) into the shaded area box, and
click the above button to obtain the z score you can use in the above equation.

9.6.6.4.2.1.2
http://davidmlane.com/hyperstat/z_table.html

10.4.1.1.2.2.1

10.4.1.1.11.1.1 Mean=35.375

10.4.1.1.11.1.2 Standard Deviation=6.304

10.4.1.1.11.2.1 Mean=35.375

10.4.1.1.11.2.2 Standard Deviation=6.304/ =1.993

10.4.1.1.11.3.1

11.6.2.9.3.1.1
Look up the z
score in a table in the back of a stats text book. To do this you will need to
subtract the probabilities above from .5000 to find the correct z score which
will give you the proportions

11.6.2.9.3.1.1.1
.25 .375 .475
.495 .4995

11.6.2.9.3.1.1.2
Look these up
in a table in the back of a sts text book to find the z scores listed below
plug oin the following probability figures .25 .125 .025 .005 .0005

11.6.2.9.3.1.2
Using the web
reference below

11.6.2.9.3.1.2.1
Plug in the following probabilities .25 .125 .025 .005 .0005 into the shaded
area of the 3^{rd} applet and click the above or below button

11.6.2.9.3.1.2.2 Web Reference

11.6.2.9.3.1.2.2.1 http://davidmlane.com/hyperstat/z_table.html

11.6.2.9.3.1.2.3
The following z
scores are associated

11.6.2.9.3.1.2.3.1
.67 1.15 1.96
2.58 3.30

11.6.2.10.1.1.1.1 Web Reference

11.6.2.10.1.1.1.1.1 http://davidmlane.com/hyperstat/z_table.html

11.10.2.2.2.1.1 The larger the sample, the smaller the standard error of the difference. (See Illustration) This illustration shows that the larger the sample size, the smaller the standard error of the mean. The same relationship is true for the standard error of a difference.

11.10.2.2.2.1.2 Some Texts [8] show you how to calculate the sample size required to reject H0. In order to do this calculation, you must make assumptions about the size of the actual difference. Many times, the size of the sample is dictated by practical consideration-time, money, or the availability of widgets.

11.10.2.2.2.2.1 Reducing the variability in the sample will produce a smaller . You can reduce variability by using reliable measuring instruments, recording data correctly, and, in short, reducing the “noise” or random error in your experiment.

12.5.16.1.1.1.1 Raw Score Formulas

12.5.16.1.1.1.1.1

12.5.16.1.1.1.2 Deviation Score Formulas

12.5.16.1.1.1.2.1

12.5.16.1.1.2.1

12.5.16.1.1.3.1

12.5.16.1.2.1.1 Raw Score Formulas

12.5.16.1.2.1.1.1

12.5.16.1.2.1.2 Deviation Score Formulas

12.5.16.1.2.1.2.1

12.5.16.1.2.2.1 =Standard error of the difference between means

12.5.16.1.2.3.1

12.6.5.19.2.1.1.1 =

12.6.5.19.5.1.1 s

12.6.5.19.5.1.1.1 Determine the standard deviation of X scores

12.6.5.19.5.1.1.2 Determine the square root of the total number of scores

12.6.5.19.5.1.1.3 Divide the product of step #1 (standard deviation of X scores) by the product of step #2 (square root of the number of X scores)

12.6.5.19.5.1.2 s

12.6.5.19.5.1.2.1 Determine the standard deviation of Y scores

12.6.5.19.5.1.2.2 Determine the square root of the total number of scores

12.6.5.19.5.1.2.3 Divide the product of step #1 (standard deviation of Y scores) by the product of step #2 (square root of the number of Y scores)

12.6.5.19.5.1.3

12.6.5.19.5.1.3.1 Square s (multiply it by itself)

12.6.5.19.5.1.3.2 Square s (multiply it by itself)

12.6.5.19.5.1.3.3 Add Squared s to Squared s

12.6.5.19.5.1.3.4 Determine the (Correlation between X & Y)

12.6.5.19.5.1.3.5 Multiply the by 2

12.6.5.19.5.1.3.6 Multiply s by s

12.6.5.19.5.1.3.7 Multiply the product of step #6 (s Xs s) by the product of step #5 ( Xs 2)

12.6.5.19.5.1.3.8 Subtract the product of step #7 (( Xs 2) Xs (s Xs s)) from the product of step #3 (Squared s + Squared s)

12.6.5.19.5.1.3.9 Obtain the square root of step #8 to obtain the score

12.6.5.19.5.2.1

12.6.5.20.2.1.1 =

12.6.5.20.2.2.1

12.6.5.20.3.1.1 Create a column with the difference between the means. That is find the difference between each pretest and posttest score (minus the posttest from the pretest) and put that number in a column

12.6.5.20.3.1.2 Create a column with the squared differences between the means. That is multiply the difference between the means by itself

12.6.5.20.3.1.3 Sum the column of squared differences (the column created in step 2)

12.6.5.20.3.1.4 Sum the column of differences (step1) and square the sum (multiply it by itself). Then divide this product by the number of score pairs.

12.6.5.20.3.1.5 Minus the product of the previous step (step 4) from the sum of the squared differences (step 3)

12.6.5.20.3.1.6 Take the number of score pairs and minus 1 from that number

12.6.5.20.3.1.7 Divide the product of step 5 by the product of step 6 to determine the () score

12.6.5.20.3.2.1 Find the difference between and

12.6.5.20.3.2.2 Obtain the square root of the number of score pairs

12.6.5.20.3.2.3 Divide by the product of step 2 to obtain the t score

12.6.5.20.3.2.4

12.7.3.7.3.1.1 =

12.7.3.7.3.6.1 Example

12.7.3.7.3.6.1.1 http://www.psychstat.smsu.edu/introbook/tdist.htm

12.7.3.7.4.1.1 Subtract the Mean of Y scores from the Mean of X scores

12.7.3.7.4.1.2 Multiply by the t score found in the table. Look across from the degrees of freedom (N-1) and under the alpha level .05. .02, .001 ect

12.7.3.7.4.1.3 Add the product of step #1 to the product of step #2 for the upper limit confidence interval

12.7.3.7.4.2.1 Subtract the Mean of Y scores from the Mean of X scores

12.7.3.7.4.2.2 Multiply by the t score found in the table. Look across from the degrees of freedom (N-1) and under the alpha level .05. .02, .001 ect

12.7.3.7.4.2.3 Subtract the product of step #1 to the product of step #2 for the lower limit confidence interval