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Class 08

Class 08. Two-tailed Tests Testing a population p. EMBS 9.1. EMBS 9.5. Measurement Scale Matters. The Roulette wheel outcomes are categorical. Start with the 904 outcomes, and create a 38-cell table of summary counts. Treat the Birth Month outcomes are categorical (J,F,M,….D) ?

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Class 08

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  1. Class 08 Two-tailed Tests Testing a population p EMBS 9.1 EMBS 9.5

  2. Measurement Scale Matters • The Roulette wheel outcomes are categorical. • Start with the 904 outcomes, and create a 38-cell table of summary counts. • Treat the Birth Month outcomes are categorical (J,F,M,….D) ? • Create a 12-cell table of summary counts. • Lorex fill outcomes are NUMERICAL. • Calculate descriptive summary statistics. • If desired, create your own bins and get a table of summary counts (histogram).

  3. MythBusters Mini-Myth play

  4. Classical Statistics EMBS p 367 • Develop and state H0 and Ha. • Specify the level of significance • α=0.05 is most common • Identify the test statistic, design and run the experiment, calculate the test statistic • 10 paired cups, double blind • Test statistic is number of correct • Calculate the p-value: the probability of observing a test statistic as “extreme” as the one calculated if H0 is true. • P(#correct ≥ 8 given she’s guessing) = .055 • P(#correct≥ 9 given she’s guessing) = .011 • P(10 correct given she’s guessing) = .001 • Reject H0 if p-value is ≤ α

  5. 1. Formulate Hypothesis H0: HA:

  6. 2. Specify the level of Significance α = 3. Identify the Test Statistic

  7. 4. Calculate the p-value: the probability of observing a test statistic as “extreme” as the one calculated if H0 is true.

  8. 5. Reject H0 if p-value is ≤ α

  9. Two-Tailed Ha • If you are looking for differences in EITHER direction • Butter affects how the toast comes down • The Lady Tasting tea can distinguish…. but not necessarily identify which is which. • The chi-squared GOF test is ALWAYS a 2-tailed test. • Differences between E and O in either direction contribute to a larger calculated chi-squared statistics

  10. Let’s Do Wunderdog as 2-tailed. • H0: He is guessing (p=.5, independent events) • Ha: He is not (p ≠.5) • Test statistic: Number correct = 87. • P( X≥87 or X≤ 62│H0 ) = 1-normdist(87,74.5,6.06,true) + normdist(62,74.5,6.06,true) = 0.020 + 0.020 = 0.04 • Conclusion: Still statistically significant at the α=0.05 level.

  11. It can be confusing • If we are more specific about what we are looking for (Ha: p>0.5), then the pvalue will be lower. • And a result is more likely to meet the 0.05 standard. • If we can’t be as specific (Ha: p≠0.5), then the pvalue will be higher. • And a result is less likely to meet the 0.05 standard.

  12. In these kinds of problems instead of using X=number correct as the test statistic….. The Sample mean (average) of the column of 149 0’s and 1’s. The Sample proportion …we can also use = X/n as the test statistic.

  13. = X/n is N(p, ) If X is N(n*p,[n*p*(1-p)]1/2) We can either compare the 87 correct to the 74.5 expected number correct Or…compare the 87/149 = 0.584 sample proportion correct to 0.5. WE JUST HAVE TO USE THE CORRECT STANDARD DEVIATION.

  14. P(60 or more in 100 if guessing) =1-BINOMDIST(59,100,.5,true) = 0.028 68% chance number correct will be within +/- 5 of 50 [100*.5*.5]^.5 =1-NORMDIST(60,50,5,true) = 0.023 [(.5*.5)/100]^.5 68% chance sample propotion will be within +/- .05 of 0.50 =1-NORMDIST(0.6,.5,.05,true) = 0.023

  15. The Standard Deviation of a sample proportion given p It is highest at p=.5 If p is 0 or 1, it is zero Requires Independent Outcomes It gets smaller as n gets bigger. It is the square root of n that matters. 4x the sample size, cuts it in half.

  16. The Standard Deviation of a sample proportion given p 68% +/- 1 * 95% +/- 2 * 95% if p=0.5 +/- Pollsters call this the margin of error

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