REVIEW: Using the t-Table (NOT the z-Table). The BODY of the t-Table actually contains the t-values, NOT the areas or probabilities like the z-Tables. As review, we locate our desired probability along the TOP row of the t-Table based on the alpha selected. NOTE that these are the UPPER-TAIL probabilities - area shown in blue. We then go down that column (under the probability) until we reach the row with the calculated degrees of freedom (df = N-1). The number at that intersection is our desired tValue. #1. WHAT are the critical t-Values with 20 df (NOT 14 as the example circles show) at the 1%, %5 and 10% levels of significance (one-tailed test - right tail). The Ho in this case would be the MEAN is GREATER THAN OR EQUAL TO ___ and the Ha: MEAN < _____ THIS WEEK'S CONCEPT: HYPOTHESIS TESTING DEALING WITH POPULATION MEANS AND THE SAMPLE MEANS AND STANDARD DEVIATIONS THAT ARE USED TO TEST THOSE POPULATION MEANS. HOWEVER, WE MUST ADJUST THE SAMPLE'S SD FOR \"SAMPLING ERROR\" and this is done by dividing the sample SD (abbreviated \"s\") by the square root of its sample size N. So, the \"adjusted\" s = original 2 2 sample s/ n. This can also be calculated as s / n. NOTE that the s is actually the VARIANCE. Remember too from Week 1 that the variance is just the square of the distance of each data point from the true mean. The \"squaring\" gets rid of negative values as you should recall. IT'S ALL ABOUT THESE DATA POINT DISTANCES FROM THE TRUE MEAN. FOR THESE HYPOTHESIS TESTS, If we KNOW the POPULATION'S STANDARD DEVIATION, we use the zTEST. LOOK AT ILLOWSKY'S SAMPLE PROBLEM 9.9 ON TEST PAGE 380 AS YOU READ ON ILLOWSKY USES THE P-VALUE TO TEST HYPOTHESES AND COMPARES THEIR CALCULATED P-VALUE TO THE ALPHA LEVEL OF SIGNIFICANCE WE CHOSE. IF THE P-VALUE IS LESS THAN THE ALPHA WE REJECT Ho. THIS IS FINE, BUT IT REQUIRES SOFTWARE TO GET THE PRECISE PROBABILITY (P-VALUE). WE WILL BE USING THE TEST STATISTIC APPROACH AND ACTUALLY READING THE TABLES. (YOU WILL SEE THAT YOU CAN STILL GET A CLOSE APPROXIMATION OF THE P-VALUE FROM THE TABLE AS WELL.). ILLOWSKY HAS AN EXAMPLE USING TEST STATISTICS ON PAGE 486 (EXAMPLE 9.11) BOTH OF THESE HYPOTHESIS TESTING APPROACHES: P-VALUE AND TEST STATISTIC WILL SUPPORT THE SAME CONCLUSION (ACCEPT OR REJECT Ho). CONFIDENCE INTERVALS CAN ALSO BE USED TO TEST HYPOTHESIS, BUT THIS APPROACH CAN (BUT NOT ALWAYS) SUPPORT DIFFERENT CONCLUSIONS. ILLOWSKY 9.9 (page 480) Baker claims his bread height is at least 15 cm. Customers say it isn't that high. This is a one-tailed test and we will use the customer's statement as our NULL Hypothesis Ho: mean bread height is less than or equal to 15 cm so Ha: mean bread height is greater than 15 cm OR Ho: < 15 cm and Ha: > 15 cm (NOTE that the Ho has the \"equals\" in it as it must) This phrasing makes it a one-tailed test to the RIGHT. If the customers had said the bread simply was NOT 15 cm, we could have ended up with a two tailed test with Ho: mean equals 15 cm and Ha: mean is simply not equal to 15 cm (height is higher or lower than 15cm). Let's keep it simple. He bakes 10 loaves that have an average height of 17cm (this is a SAMPLE) The Baker has produced enough loaves of bread (his POPULATION of bread) over his career to have calculated a standard deviation = 0.5 and we adjust this for \"error\" as well so it is 0.5/N and N = 10 so = 0.5/10 = 0.16 NOW, since we have the standard deviation we can use the z-Test statistic. The 10 loaf sample mean is 17 cm, this is our x-value we are using to test the accuracy of the Population mean of 15 cm. We simply standardize that x-value: z-test statistic = (x - mean) /adjusted SD = (17 - 15) / 0.16 = 12.5 This is the z-TEST value we now go to our +z-Table with. You will quickly see that a z-value of +12.5 is WAY off the chart with more than 99.999% to the LEFT ( and less than 0.00000000. . .1 to the right as the PROBABILITY. Normally you would compare this calculated z-Test statistic to our critical values, but you can see that at whatever level of significance we chose (1%, 5% or 10%) we are MUCH further to the right (in the \"reject Ho) area than any of them. So, we REJECT Ho and accept Ha: This Baker's bread has an average height of greater than 15 cm (we don't say equal to or greater than since the Ho included the equals and rejected all of it). NO POPULATION STANDARD DEVIATION MORE OFTEN we do NOT know the POPULATION'S SD, so we substitute the SAMPLE'S STANDARD DEVIATION and use the t-TEST. LOOK AT ILLOWSKY EXAMPLE 9.16 ON PAGE 488 THAT USES THE t-TEST STATISTIC. THEY STILL USE THE PVALUE BUT WE WILL CALCULATE THE TEST STATISTIC This example deals with scores on a Statistics test. Student believe the mean score is 65 and the instructor believes it is higher. So, let's go with Ho: mean < 65 and Ha: > 65 We don't have POPULATION mean of SD. So we use a sample. This is a \"single population mean\" one-tailed test to the right (since the Ha: has the \">\") The chosen level of significance is 5%, but we can't use the critical z-values we tabulated earlier. We need to find the critical t-Value. Ten student tests were sampled so N= 10 and the df = 10 - 1 = 9 The sample of the 10 tests had a mean (X-bar as it's referred to) of 67. And we need to calculate the sample's standard deviation (s) which ends up s = 3.20 and then correct it for sampling error = s/10 = 3.197/3.162 = 1.011 Now, our t-Test statistic = (X-bar - mean)/ adjusted SD = (67 - 65)/1.01 = 1.978 WHAT IS THE CRITICAL t-VALUE WITH AN ALPHA OF 5% AND A df = 9 ? Go to the T-Table and find 5% (0.05) along the TOP row. Go down that COLUMN till you come to the row for a DF of 9 and read: 1.833. THIS IS OUR CRITICAL t-VALUE FOR THIS PROBLEM. COMPARE THE CALCULATED t-TEST STATISTIC 1.978 TO THE CRITICAL VALUE 1.833 AND WE SEE THAT THE TEST STATISTIC IS GREATER THAN (FURTHER TO THE RIGHT) OF THE CRITICAL VALUE, HENCE WE REJECT Ho. THE MEAN IS GREATER THAN 65. ILLOWSKY USED SOFTWARE BUT LET'S GO INTO THE t-TABLE WITH OUR t-TEST STATISTIC CALUCLATED VALUE OF 1.978 AT A df OF 9. FIND IT? NOW, LOOK UP TO THE TOP ROW OF THE TABLE AND WHAT PROBABILITIES ARE WE BETWEEN? WE ARE LESS THAN 0.0500 BUT MORE THAN 0.025. PER ILLOWSKY, THE ACTUAL P-VALUE IS 0.0396. SEE HOW YOU CAN GET AN APPROXIMATE P-VALUE JUST FROM THE TABLE? HOWEVER, WE WILL USE THE TEST STATISTIC APPROACH. YOU CAN CERTAINLY PRACTICE USING THE P-VALUE SOFTWARE, BUT SUBMIT ANSWERS USING TEST STATISTICS. WE CAN ALSO TEST PROPORTIONS RATHER THAN MEANS. FOREXAMPLE WE BELIEVE THAT 65% OF STUDENTS GET A \"C\" OR BEETER IN STATSTICS. THERE IS NO MEAN PROVIDED. WE GET WHAT WE NEED FROM A SAMPLE. CHECK ILLOWSKY EXAMPLE 9.17 PAGE 489) ONE LAST HYPOTHESIS INVOLVES COMPARING TWO MEANS AS IN OUR DRINKING OR NOT DRINKING COFFEE BEFORE A LONG LECTURE. THE Ho: would be that the means are equal expressed as Ho: M1 - M2 = 0 and the Ha: could be a < or > depending on how it's stated. Illowsky pages 530-531 have the formulas for t-test statistic calculation and the degrees of freedom (df) calculation FOR THESE MEAN COMPARISON HYPOTHESIS TESTS. Both are a little involved, but you NEED TO USE THEM (NOT software) to calculate these necessary values. Round off your calculated df to the nearest whole number and then YOU CAN AND NEED TO USE A TABLE TO FIND THESE VALUES. HYPOTHESIS TESTING CAN MAKE MISTAKES. THESE ARE REFERRED TO AS TYPE I AND TYPE II ERRORS. WE COULD REJECT Ho WHEN IT IS ACTUALLY TRUE OR ACCEPT Ho WHEN IT IS ACTUALLY FALSE. BIG MISTAKES LEGALLY. ACTION Do NOT reject Ho Reject H0 Ho IS ACTUALLY TRUE FALSE Correct Outcome Type II Error Type I Error Correct Outcome 2. You choose an alpha level of .01 and then analyze your data. a. What is the probability that you will make a Type I error given that the null hypothesis is true? b. What is the probability that you will make a Type I error given that the null hypothesis is false? 3. You are conducting a study to see if students do better when they study all at once or in intervals. One group of 12 participants took a test after studying for one hour continuously. The other group of 12 participants took a test after studying for three twenty minute sessions. The first group had a mean score of 75 and a variance of 120. The second group had a mean score of 86 and a variance of 100. a. What is the calculated t value? Are the mean test scores of these two groups significantly different at the .05 level? b. What would the t value be if there were only 6 participants in each group? Would the scores be significant at the .05 level? 4. List 3 measures one can take to increase the power of an experiment. Explain why your measures result in greater power. 5. . A Nissan Motor Corporation advertisement read, \"The average man's I.Q. is 107. The average brown trout's I.Q. is 4. So why can't man catch brown trout?\" Suppose you believe that the brown trout's mean I.Q. is greater than four. You catch 12 brown trout. A fish psychologist determines the I.Q.s as follows: 5; 4; 7; 3; 6; 4; 5; 3; 6; 3; 8; 5. Conduct a hypothesis test of your belief. : SHOW ALL STEPS FROM STATING THE HYPOTHISIS TO DRAWING THE CONCLUSION. MAKE SURE TO PROVIDE YOU CALCULATED TEST STATISTIC AND THE CRITICAL VALUE. 6. . At Rachel's 11th birthday party, eight girls were timed to see how long (in seconds) they could hold their breath in a relaxed position. After a two-minute rest, they timed themselves while jumping. The girls thought that the mean difference between their jumping and relaxed times would be zero. Test their hypothesis. Relaxed time (seconds) Jumping time (seconds) 26 21 47 40 30 28 22 21 23 25 45 43 37 35 29 32 . USE THE TEST STATISTIC APPROACH USING THE EQUATIONS ON PAGES 530-531 IN ILLOWSKI FOR THE tTEST STATISTIC AND THE DEGREES OF FREEDOM (THE LATTER TO GET THE CRITICAL t-VALUE. LANE ALSO HAS PERHAPS A SIMPLER VERSION OF THESE EQUATIONS. SHOW ALL STEPS