Binomial Distribution and Applications Binomial Probability Distribution n
Binomial Distribution and Applications
Binomial Probability Distribution n
Binomial Distribution n
Binomial Distribution n
Binomial Distribution n SSF n SFS FSS
Binomial Distribution n
Binomial Distribution n
Binomial Table in JMP (linked below Ch 3 notes) Right-click at the top of these columns, click on Formula and change the number of trials (n) to the value you want. Do the same for the probability of success (pi).
Binomial Table in JMP (linked below Ch 3 notes)
Mean, Variance, and Standard Deviation of a Binomial Random Variable n
Mean and Variance of a Binomial Random Variable n
Example: 100 flips of a fair coin n
Example: 100 flips of a fair coin n
Mean and Variance of a Binomial Random Variable n
Empirical Rule with Binomial n
Empirical Rule with Binomial n
Example: Treatment of Kidney Cancer n
Example: Treatment of Kidney Cancer n
n Example: Treatment of Kidney Cancer
n Example: Treatment of Kidney Cancer
Example: Treatment of Kidney Cancer
Results and “The Question” n Now suppose that using the new treatment we find that 16 out of the 40 patients survive at least 5 -years past diagnosis. n The Question: Does this result suggest that the new therapy has a better 5 -year survival rate than the current treatments, i. e. is the probability that a patient survives at least 5 years greater than. 20 or a 20% chance when treated using the new therapy?
What do we consider in answering the question of interest? We essentially ask ourselves the following: n If we assume that new therapy is no better than the current treatments, what is the probability we would see these results by chance variation alone? n More specifically what is the probability of seeing 16 or more successes out of 40 if the success rate of the new therapy is. 20 or 20% as well, i. e. it is no better than current treatments?
Connection to Binomial n
Example: Treatment of Kidney Cancer n n
Example: Treatment of Kidney Cancer n n n
Example: Treatment of Kidney Cancer n
Example: Treatment of Kidney Cancer n
Example: Treatment of Kidney Cancer n
Conclusion n
Conclusion n
Conclusion n
Conclusion n
n Example: 2020 Presidential Election
n Example: 2020 Presidential Election
n Example: 2020 Presidential Election
n Example: 2020 Presidential Election
n Example: 2020 Presidential Election
n Example: 2020 Presidential Election
n Example: 2020 Presidential Election
n Example: 2020 Presidential Election
n Example: 2020 Presidential Election 95% 5%
Discrimination in the Workplace Internal records show that 60% of employees at Walmart who are eligible for their management training program are women. Employees completing the program are promoted to assistant store or department managers, and thus receive a substantial raise. In a random sample of 50 individuals that had been recently selected for participation in the management training program, 21 of them were women. Does provide evidence of bias against women in the selection process?
Discrimination in the Workplace Question: Does this provide evidence that the selection was not truly “random” and there are too few women selected for the training program, suggesting possible sex bias? Evidence: 21 of 50 employees, 21/50 =. 42 or 42%, recently selected for the management training program were women. As 60% of employees eligible for the program are women, there is underrepresentation of women in the sample. However, this could just be
n Discrimination in the Workplace
n Discrimination in the Workplace
n Discrimination in the Workplace
Sign Test for Paired Differences Patient Before (B) Infusion After (A) Infusion Differenc e (A – B) 1 2 1 -1 16 0 0 0 2 0 0 0 17 0 3 3 3 0 0 0 18 2 3 1 4 1 0 -1 19 2 3 1 5 3 3 0 20 3 2 -1 6 1 1 0 21 0 4 4 7 1 3 2 22 0 3 3 8 0 0 0 23 1 2 1 9 0 0 0 24 0 3 3 10 1 0 -1 25 0 2 2 11 1 1 0 26 1 1 0 12 1 1 0 27 3 3 0 13 2 1 -1 28 1 2 1 14 3 1 -2 29 0 2 2
Sign Test for Paired Differences n It can be used when the response is ordinal. n Best used when the response is difficult to quantify and only improvement can be measured, i. e. subject got better, got worse, or no change. n Magnitude of the paired difference is lost when using this test. n We will consider two other tests that are used more commonly later in the course (dependent samples t-test and the Wilcoxon signed-rank test. ) n
Sign Test for Paired Differences A study evaluated hepatic arterial infusion of floxuridine and cisplatin for the treatment of liver metastases of colorectral cancer. n Performance scores for 29 patients were recorded before and after infusion. This is sometimes referred to as pre-test vs. posttest. n Is there evidence that patients had a better performance score after infusion? n
Example: Sign Test Patient Before (B) Infusion After (A) Infusion Differenc e (A – B) 1 2 1 -1 16 0 0 0 2 0 0 0 17 0 3 3 3 0 0 0 18 2 3 1 4 1 0 -1 19 2 3 1 5 3 3 0 20 3 2 -1 6 1 1 0 21 0 4 4 7 1 3 2 22 0 3 3 8 0 0 0 23 1 2 1 9 0 0 0 24 0 3 3 10 1 0 -1 25 0 2 2 11 1 1 0 26 1 1 0 12 1 1 0 27 3 3 0 13 2 1 -1 28 1 2 1 14 3 1 -2 29 0 2 2
Sign Test The sign test looks at the number of (+) and (-) differences amongst the nonzero paired differences. n A preponderance of +’s or –’s can indicate that some type of change has occurred. n If in reality there is no change as a result of infusion we expect +’s and –’s to be equally likely to occur, i. e. P(+) = P(-) = . 50 and the number of each observed follows a binomial distribution. n
Example: Sign Test n Given these results do we have evidence that performance scores of patients generally improves following infusion? n Need to look at how likely the observed results are to be produced by chance variation alone.
Example: Sign Test 17 nonzeros differences, 11 +’s 6 –’s Before (B) Infusion After (A) Infusion Patient Before (B) Infusion After (A) Infusion Differenc e (A – B) 1 2 1 -1 16 0 0 0 2 0 0 0 17 0 3 3 3 0 0 0 18 2 3 1 4 1 0 -1 19 2 3 1 5 3 3 0 20 3 2 -1 6 1 1 0 21 0 4 4 7 1 3 2 22 0 3 3 8 0 0 0 23 1 2 1 9 0 0 0 24 0 3 3 10 1 0 -1 25 0 2 2 11 1 1 0 26 1 1 0 12 1 1 0 27 3 3 0 13 2 1 -1 28 1 2 1 14 3 1 -2 29 0 2 2 Patient Differenc e (A – B) - + + +
Example: Sign Test n
Example 2: Sign Test Resting Energy Expenditure (REE) for Patient with Cystic Fibrosis n A researcher believes that patients with cystic fibrosis (CF) expend greater energy during resting than those without CF. To obtain a fair comparison she matches 13 patients with CF to 13 patients without CF on the basis of age, sex, height, and weight. She then measured there REE for each pair of subjects and compared the results.
Example 2: Sign Test There are 11 +’s & 2 –’s out of n = 13 paired differences.
Example 2: Sign Test
Example 3: Sign Test for the Median n
Example 3: Sign Test for the Median n
Example 3: Sign Test for the Median n
Example 3: Sign Test for the Median n Duluth
Example 3: Sign Test for the Median Duluth
Example 3: Sign Test for the Median Notice the sample median is. 625 ppm, which is indeed greater than. 57 ppm. However, even if the median were right at. 57 ppm, this result could be seen by chance variation alone. Inspection of the raw data collected shows there were no walleyes with a Hg level of. 57 ppm, so when subtracting. 57 from each observation, there are no zero differences. Using the Local Data Filter and highlight observations where the Hg level is greater than. 57 ppm, we see that 32 of the fish sampled had a Hg levels above. 57. Thus 26 of the fish had Hg levels below. 57 ppm.
Example 3: Sign Test for the Median
Example 3: Sign Test for the Median
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