1. Copyright © Texas Education Agency, 2012. All rights reserved. Hypothesis Testing Within One Set of Numbers Statistics & Risk Management 1

2. Copyright © Texas Education Agency, 2012. All rights reserved. 2 “Copyright and Terms of Service Copyright © Texas Education Agency. The materials found on this website are copyrighted © and trademarked ™ as the property of the Texas Education Agency and may not be reproduced without the express written permission of the Texas Education Agency, except under the following conditions: 1)Texas public school districts, charter schools, and Education Service Centers may reproduce and use copies of the Materials and Related Materials for the districts’ and schools’ educational use without obtaining permission from the Texas Education Agency; 2) Residents of the state of Texas may reproduce and use copies of the Materials and Related Materials for individual personal use only without obtaining written permission of the Texas Education Agency; 3) Any portion reproduced must be reproduced in its entirety and remain unedited, unaltered and unchanged in any way; 4) No monetary charge can be made for the reproduced materials or any document containing them; however, a reasonable charge to cover only the cost of reproduction and distribution may be charged. Private entities or persons located in Texas that are not Texas public school districts or Texas charter schools or any entity, whether public or private, educational or non-educational, located outside the state of Texas MUST obtain written approval from the Texas Education Agency and will be required to enter into a license agreement that may involve the payment of a licensing fee or a royalty fee. Call TEA Copyrights with any questions you have.

3. Copyright © Texas Education Agency, 2012. All rights reserved. Comparison Within We need a way to measure the difference between obtained results and what would occur randomly. We need a way to measure the difference between obtained results and what would occur randomly. For example: At the Mall of America we asked everyone their age as they entered the store for one hour on Saturday. For example: At the Mall of America we asked everyone their age as they entered the store for one hour on Saturday. We discarded any ages under 16 or over 65 (trimmed). We collected a sample of 100 useable responses. We grouped them in 10 year blocks. We discarded any ages under 16 or over 65 (trimmed). We collected a sample of 100 useable responses. We grouped them in 10 year blocks. 3

4. Copyright © Texas Education Agency, 2012. All rights reserved. Our null-hypothesis is that we will get an equal number of people in each of the five age groups. Our null-hypothesis is that we will get an equal number of people in each of the five age groups. H 0 : P 1 = P 2 = P 3 = P 4 = P 5 4 Comparison Within

5. Copyright © Texas Education Agency, 2012. All rights reserved. AgeExpectActualDiffDiff 2 Diff 2 /Expect 16-252013-7492.45 26-3520288643.20 36-452017-39.45 46-5520277492.45 56-652015-5251.25 Totals100 0196* 9.80 The 9.80 is the chi-square test value. Degrees of Freedom ( df ) = Groups – 1 = 5 – 1 = 4 We will use a Type I (alpha) Error Rate of.05 5 Comparison Within

6. Copyright © Texas Education Agency, 2012. All rights reserved. df.10.05.01 df.10.05.01 12.713.846.641623.5426.3032.00 24.005.999.211724.7727.6033.41 36.257.8211.341825.9928.8734.80 47.789.4913.281927.2030.1436.19 59.2411.0715.092028.4131.4137.57 610.6412.5916.812129.6232.6738.93 712.0214.0718.482230.8133.9240.29 813.3615.5120.092332.0135.1741.64 914.6816.9221.672433.2036.4242.98 1016.9918.3123.212534.3837.6544.81 1117.2819.6824.722635.5638.8845.64 1218.6521.0326.222736.7440.1149.96 1319.8122.3627.672837.9241.3448.28 1421.0623.6829.142939.0952.5649.59 1522.3125.0030.583040.2643.7750.89 Critical Values for Chi-Squared Test 6

7. Copyright © Texas Education Agency, 2012. All rights reserved. What Does This Mean? We found that the Chi-Squared value of 9.80 is larger than the Critical Value of 9.49 We found that the Chi-Squared value of 9.80 is larger than the Critical Value of 9.49 So we will reject the Null-Hypothesis So we will reject the Null-Hypothesis – Because there is a significant difference between the obtained results and what would be expected as a result of just random responses. Notice how close that was! 7

8. Copyright © Texas Education Agency, 2012. All rights reserved. What Does That Mean? That there is a significant difference of ages in the people we interviewed beyond what we would expect (Probability). That there is a significant difference of ages in the people we interviewed beyond what we would expect (Probability). 8

9. Copyright © Texas Education Agency, 2012. All rights reserved. Chi-Square Formula? Chi-square value Obtained Score(s) Expected Score(s) 9

10. Copyright © Texas Education Agency, 2012. All rights reserved. Excel Formula =CHITEST(B1:B5,C1:C5) This function uses two lists of Expected (column B) and Observed (Column C) 10