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Probability & Statistics I
Robb Sinn
Math 3350 / 6350 NGCSU, Fall 2006
Hypothesis Testing: More Exam-
ples
Other Hypothesis Tests
A near inÖnitude of statistical tests exist, and each advantages and disad- vantages. The point here is not to go to deeply into any one type of test but demonstrate several. The hypothesis testing steps are always roughly the same.
Steps in Hypothesis Testing
- Select appropriate statistical test.
- Set up the null and research hypotheses.
- Verify the assumptions. Proceed with steps 3 - 11 only if assumptions are valid.
- Select an appopriate based on power vs. Type I error percentages, sample size, main e§ect vs. side e§ects and other research considerations. (Power can be di¢ cult to estimate a priori)
- Based on , compute the critical value.
- Compute the test statistic (or p-value).
- Compare the test statistic to the critical value (or p to ).
- Either reject or fail to reject the null.
- State the inference in the real-world setting of the research.
- For ANOVA, perform post hoc analyses to determine signiÖcant pairwise di§erences (Tukey HSD) or contrasts (Sche§e).
- Consider performing a post hoc estimate of power, especially in pilot tests or when doing exploratory research.
We will look at the following examples: the t-interval which is used to es- timate an unknown population mean, the Chi-Square "Goodness of Fit" test which tests whether population proportion conform to a given probability model, the Chi-Square "Test of Independence" which uses a 2-way table to determine if two categorical variables are dependent or indepent. The Önal branch of basic stastical tests will follow in another section: regression or correlation studies.
t-Intervals
Recall the t-test is a more robust version of the z-test. The t-interval is likewise the more robust version of the z-interval. The z-interval is contructed in exactly the same way, but it is less useful. The idea is to take a data set whose characteristics (determined by data analysis: histograms, stem plots, various checks for outliers) make it likely to be a sample drawn from an approximately normal distribution.
Example 1: t-Interval College students were asked to report their weekly TV watching hours and the hours they spent using a computer. This is a paired samples or depen- dent samples design. The unit of study is the di§erence in TV watching vs. computer use. The data is given below:
Student 1 2 3 4 5 6 7 8 9 10 11 12 13 Computer 30 20 10 10 10 0 35 20 2 5 10 4 50 TV 2 2 14 2 6 20 14 1 14 10 15 2 10 Di§erence 28 18 4 8 4 20 21 19 12 5 5 2 40
Student 14 15 16 17 18 19 20 21 22 23 24 25 Computer 5 8 30 40 15 40 3 21 2 9 14 21 TV 6 20 20 35 15 5 13 35 1 4 0 14 Di§erence 1 12 10 5 0 35 10 : 5 14 1 5 14 7
Estimate the mean with 95% conÖdence.
The following data table can be veriÖed:
xd = 5: 4 s = 15: 2 n = 25
The same assumptions apply: normality, independence and homogeneity of the variance. Normality is veriÖed since n 25 , so the Law of Large Numbers takes over. How do we check independence of the data points? Well, with common sense. Two or more scores might be dependent if roommates were in the sample. Then, the TV watching of one might ináuence the other.
reverse of calculating the critical value. We are now constructing a theoretical distribution around the sample mean and comparing it to the null hypothetical population mean. Note that the test statistic is given by
t = (^) pxsd n
and if we multiply both sides by psn , we have
xd t psn.
With the correct value for t, we can estimate d. The calculations are as before except that now we Önd = 0: 1 in the two-tailed t-test table: t^ = 1: 71. When we decrease the conÖdence %, we decrease the range of the interval. More values for d will fall outside this interval, so it can be smaller.
5 : 4 1 : 71 15 p 24 :^2! (0: 0944 ; 10 :706)
Now the 90% conÖdence interval does NOT include the null hypothesized d = 0, so we reject the null. Our probability-based estimate is that, at least 95% of the time, d > 0 : 0944 because 90% of the time d 2 (0: 0944 ; 10 :706) and 5% of the time d > 10 : 706. This indicates that it is statistically unlikely that d = 0. We can also compute the test statistic t for the dependent samples test and compare it to the one-directional critical value t^ for = 0: 05 with 24 degrees of freedom (see above): t^ = 1: 711. Then, we compare
t = (^) pxsd n = (^15) p^5 :^4 : 2 24
= 1: 740 4 > 1 :711 = t
and so reject the null. Most of the uses of conÖdence intervals are estimates of a statistic, often a di§erence in means such as the post hoc Tukey HSD intervals (see Hypothesis Testing, Part 1).
^2 Goodness of Fit
The goodness of Öt chi-square test measures how close to a probability model actual data has conformed. This test is often used to analyze genetics. One hypothesizes dominant and recessive traits and produces the proportions of traits the o§spring should have. Then the actual o§springsítraits are tabulated and compared the model. The example problem below is more interesting to read knowing that I used it in my NGCSU interview when I was teaching a
stats class. I had underestimated how conservative North Georgia is, and this example went like a pregnant polevaulter.
Example 2: Joker and Judy The notorious river boat gambler Joker McGants was plying his trade a bit too well in the ATL one April night. Sitting amidst a pile of chips, bills and booze, Joker realized that Bugsy Haygood was downright ill-tempered, ill- mannered and ill-equipped to restrain his erupting anger. ìYouíre using loaded dice, mug,î Bugsy accused. Joker donned a look so innocent it was worthy of the seraphim, as well it should have been. Joker had practiced in front of a mirror for hours in preparation for just such occasions. Given that Bugsyís stock and trade involved 9mm projectiles and somebody named Soprano, the sweat that rolled icily down Jokerís spine was understand- able. ìYouíre welcome to perform any test you wish on these dice, sir,î Joker agreed.Bugsy performed the various test gamblers use to ascertain that dice are true. No shaved corners, no weighted imbalances, no signs of any alterations. Joker smiled his winsome best. Bugsy was not amused. Just as Joker was judging the possibilities of a live-to-gamble-another-day escape, the cocktail waitress Jody DeLong o§ered a suggestion. ìIím taking a statistics course at North Georgia, and I know a way to tell if those dice are loaded. We can use chi-square.î Both Joker and Bugsy agreed, although a debate quickly ensued. ìI prefer using = 0: 01 ,î Joker suggested. ìDonít want any Type I errors, you know.î ìI disagree,îBugsy told him. ìI think an = 0.1 would better control that insidious Type II error. We do want as much power as possible, donít we?î Jody did not like the evil glint in Bugsyís eye nor the way his hand inched toward the coat pocket that bulged noticeably. Jody, as bright as she was beautiful, just rolled her eyes, ìEveryone knows that has to be .05. Duh.î The die was cast 60 times with the following, spine tingling results:
Outcome 1 2 3 4 5 6 Frequency 14 12 9 8 8 9
The probability model should be clear: for fair dice we would expect 10 of each outcome in 60 rolls. So the null and alternative hypothesis are
H 0 : p 1 = p 2 = p 3 = p 4 = p 5 = p 6 = (^16) Ha : At least one pi 6 = 16 ; 1 i 6
Yes No Totals % Males 26 35 61 30.81% Females (^) 86 51 137 69.19% Totals (^) 112 86 198 % (^) 56.57% 43.43%
Yes No Men 26 35 Women 86 51
Test these two variables for indepence at the = 0: 05 level of signiÖcance.
The hypothesis is straightforward:
H 0 : Variables are independent Ha : Variables are dependent
For this test, we have to work a bit harder to construct the "expected" outcome matrix. A spreadsheet like MS Excel can quickly generate the needed tallies.
Since about 31% of the sample are males, and since about 57% of respondents said "Yes," we expect the count in "Males:Yes" cell to be 198(:3081)(:5657) = 34 : 510 , which we will round to 35. If the responses are independent of gender, we expect about 35 males to respond with "Yes." The same calculations are used for each of the other cells, so the matrix of "expected" outcomes is shown below:
Yes No Men 35 26 Women 77 60
We compare cell by cell as follows:
^2 = (26 35)
2 35 +^
(35 26)^2 26 +^
(86 77)^2 77 +^
(51 60)^2 60
^2 = 7: 8316
The degrees of freedom for two-way tables is (r 1)(c 1) where the observed matrix has r rows and c columns. Here, we have one degree of freedom, so the critical value ^2 ^ is found using in the table [?? ]: 3 : 841. Since
^2 = 7: 8316 > 3 :841 = ^2
we reject the null hypothesis that these two variables are unrelated. The answer choice of Yes vs. No depends signiÖcantly upon the gender of the person answering the question, so yes, men who lack movie star looks can get by with a good personality. Women face a less friendly dating landscape, it would appear.
Example 4: Are GPA and Seat-Belt Use Related? I would restate the question in a more incendiary fashion: are dumb people less likely to use life-saving equipment like seat belts? Of course, there is much more to the GPA metric (especially in high school) than intelligence, but it makes the example more fun. Part of the 2001 Youth Risk Behavior Assessment asked seniors in high school about their use of seat belts. The data is presentend in the two-way table below. Given the large sample size, it would be appropriate to test this hypothesis at a conservative level of signiÖcance, so set = 0: 01.
Observed A/B C D/F Usually/Always 1354 428 65 Rarely/Never 180 125 40
The hypothesis is always similar:
H 0 : GPA and seat-belt use are independent Ha : GPA and seat-belt use are dependent
The hardest part of a the 2-way table test is calculating the "expected" matrix. Again, Excel will make the calculations easier. We compute row and column percentages, then recompute each cell based on them.
The new "expected" matrix is given below.
