1. The T-Test Are our results reliable enough to support a conclusion?

2. Imagine we chose two children at random from two class rooms… 12 … and compare their height …

3. 12 … we find that one pupil is taller than the other WHY?

4. REASON 1: There is a significant difference between the two groups, and pupils in room 2 are actually taller than pupils in room 1. 1 YEAR 7 2 YEAR 11 maybe kids in room B are older than kids in room A

5. REASON 2: By chance, we picked a short pupil from room 1 and a tall one from room 2 12 Ciera (Year 11) Dominic (Year 11)

6. How do we decide which reason is most likely? MEASURE MORE STUDENTS!!!

7. If there is a significant difference between the two groups… 12 … the mean height of the two groups should be very… … DIFFERENT

8. If there is no significant difference between the two groups… 12 … the mean height of the two groups should be very… … SIMILAR

9. Remember: Living things normally show a lot of variation, so…

10. It is VERY unlikely that the mean height of our two samples will be exactly the same Room 1 Average height = 162 cm Room 2 Average height = 168 cm Is the difference in average height of the samples large enough to be significant?

11. Student’s t-test The Student’s t-test is a statistical test that compares the averages and standard deviations of two samples to see if there is a significant difference between them. We start by calculating a number, t t can be calculated using the equation: ( x 1 – x 2 ) (s 1 ) 2 n1n1 (s 2 ) 2 n2n2 + t = Where: x 1 is the mean of sample 1 s 1 is the standard deviation of sample 1 n 1 is the sample size of sample 1 x 2 is the mean of sample 2 s 2 is the standard deviation of sample 2 n 2 is the sample size in sample 2

12. Worked Example: Random samples were taken of pupils in room 1 and room 2. Their recorded heights are shown below… Students in Room 1Students in Room 2 Student Height (cm) 145140138142154148153157161162 154158160166 162163167172

13. Students in Room 1Students in Room 2 Student Height (cm) 145140138142154148153157161162 154158160166 162163167172 Step 1: Calculate the mean height for each sample 152.3 cm Room 1: x 1 = 161.7 cm Room 2: x 2 =

14. Step 2: Find the absolute value of the difference between the means 9.4x 2 – x 1 = 161.7 – 152.3 =

15. Step 3: Work out the standard deviation for each sample Room 1: s 1 =Room 2: s 2 = 10.48 7.66

16. Step 4: Square the standard deviation for each group Room 1: s 1 2 = Room 2: s 2 2 = 109.83 58.68

17. Room 1: 109.80 ÷ 10 = 10.98 Room 2: 58.68 ÷ 10 = Step 5:Divide each squared standard deviations by the sample size of that group. 5.87

18. 10.98 + 5.87= 16.85 Step 6: Add these two values.

19. √16.85 = 4.10 Step 7: Take the square root of the number

20. 9.4 ÷ 4.10 = 2.28 Step 8: divide the difference in the means (step 2) by the standard error of the difference (step 7) “t”

21. Step 9: determine the degrees of freedom (df) for the test. In the t-test, the degrees of freedom is the sum of the sample sizes of both groups minus 2. (10 + 10) - 2 = 18 “df”

22. Use the 95% (p=0.05) confidence limit Step 10: Given the df, look up the critical t-value in a standard table of significance df0.2.01.05.02.01.002 1 3.0786.31412.70631.82163.657636.619 21.8862.9204.3036.9659.92531.598 3 1.6382.3533.1824.5415.84112.941 41.5332.1322.7763.7474.6048.610 5 1.4762.0152.5713.3654.0326.859 61.4401.9432.4473.1433.7075.959 7 1.4151.8952.3652.9983.4995.405 81.3971.8602.3062.8963.3555.041 9 1.3831.8332.2622.8213.2504.781 101.3721.8122.2282.7643.1694.587 11 1.3631.7962.2012.7183.1064.437 121.3561.7822.1792.6813.0554.318 13 1.3501.7712.1602.6503.0124.221 141.3451.7612.1452.6242.9774.140 15 1.3411.7532.1312.6022.9474.073 161.3371.7462.1202.5832.9214.015 17 1.3331.7402.1102.5672.8983.965 181.330 1.734 2.1012.5522.8783.922 19 1.3281.7292.0932.5392.8613.883 201.3251.7252.0862.5282.8453.850 21 1.3231.7212.0802.5182.8313.819 221.3211.7172.0742.5082.8193.792 23 1.3191.7142.0692.5002.8073.767 241.3181.7112.0642.4922.7973.745 25 1.3161.7082.0602.4852.7873.725 261.3151.7062.0562.4792.7793.707 27 1.3141.7032.0522.4732.7713.690 281.3131.7012.0482.4672.7633.674 29 1.3111.6992.0452.4622.7563.659 301.3101.6972.0422.4572.7503.646 40 1.3031.6842.0212.4232.7043.551 601.2961.6712.0002.3902.6603.460 120 1.2891.6581.9802.3582.6173.373  1.2821.6451.9602.3262.5763.291

23. SO WHAT? If your calculated t value is less than the number in the table, you conclude that the difference between the means for the two groups is NOT significantly different. If your calculated t value is greater than the number in the table, you conclude that the difference between the means for the two groups is significantly different.

24. Calculated t-value = 2.28 Critical t- value = 2.101 Our calculated value of t is above the critical value, therefore, there is a significant difference between the height of students in samples from room 1 and Room 2

25. Do not worry if you do not understand how or why the test works Follow the instructions CAREFULLY

26. T-test in Excel Excel calculates a T-test in a slightly different way. Rather than giving you the t value and comparing it to a table, Excel simply tells you the probability that the means are different simply due to chance. This is called a “P value.”

27. T-test in Excel Follow these steps to calculate a P value using a t-test with Excel: –Step 1: Create two columns for the data of interest. –Step 2: Click on a blank cell where you wish the P value to appear. –Step 3: Then click “fx” on the Excel toolbar and choose “statistical” from the “function” list, then “TTest” from the list. –Step 4: Set the t-test parameters: For “Array1” highlight the data from one sample; for “Array2”, highlight the data in the second column. Enter “2” in the box for “Tails.” Lastly, you will have to select the “Type” of t-test. or our purposes type “2.” After answering these questions click “OK” and the P value will appear. The P value will fall between zero and one.

28. T-test in Excel What does my P value mean? –a P value of.05 or less is significant difference in the means of the two groups. –a P value of.06 or more is NOT a significant difference in the means of the two groups; the difference is due to chance!

29. The End!