Confidence interval for difference in population

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Confidence interval for difference in population proportions: ( ˆ p 1 - ˆ p 2 ) ± z α/ 2 · s ˆ p 1 (1 - ˆ p 1 ) n 1 + ˆ p 2 (1 - ˆ p 2 ) n 2 Example: A study on side-effects of a new medication is being con- ducted. 80 people are given a placebo, and 120 people are given the new medication. Of those receiving a placebo, 10 have dry mouth. Of those receiving the medication, 18 have dry mouth. Let p 1 denote the proportion of placebo patients who have dry mouth, and p 2 denote the proportion of medicated patients who have dry mouth. Find a 90% confidence interval for p 1 - p 2 . 23
Hypothesis tests on the difference in population proportions: We use the following as our test statistic. z = ( ˆ p 1 - ˆ p 2 ) - ( p 1 - p 2 ) r ˆ p 1 (1 - ˆ p 1 ) n 1 + ˆ p 2 (1 - ˆ p 2 ) n 2 Example: Is there evidence that the dry-mouth proportion of those on the medication is larger than the dry-mouth proportion for those on the placebo? 24
Example: Researchers wish to investigate the impact of nutritional coun- seling on birth weight for first-time mothers. Two groups were formed, with one group receiving nutritional advising. The table below shows the number of underweight babies and sample sizes for the two groups. Is there significant evidence to show that advising reduced the proportion of underweight babies by at least 4 percentage points? Test at the level α = 0 . 05 . No Advising Advising Underweight 29 24 Total 200 300 25
Example: Using this as a pilot study, estimate the common sample size needed to create a 95% confidence interval no more than 2% away from the true difference in proportions. Hint: Let n 1 = n 2 = n . 26
Sets 25 and 26 In the next two sets, we will deal with comparing the means of two samples from two populations. Parameter of interest: μ 1 - μ 2 , the difference in the means. Point estimate: x 1 - x 2 . In the handout, (“Test Statistics & Confidence Intervals (from two inde- pendent samples)” on the course page), there are four different scenarios. The first scenario requires σ 1 and σ 2 to be known; this scenario is highly unlikely to occur, and is presented only for completeness purposes. The second scenario has no restrictions, other than a large sample size ( 40 or more) is taken independently from two populations. 27
Example: The Hamilton Depression Scale (HAM-D) is used by psychia- trists to assess the severity of depression in patients, on a scale from 0 to 54 . Two groups of patients are selected for study, and randomly assigned to antidepressant medication A or B . Of the 80 patients taking medication A , their HAM-D rating decreased by an average of 7 . 6 points, with a standard deviation of 1 . 3 . Of the 100 patients taking medication B , their HAM-D rating decreased by an average of 8 . 1 points, with a standard deviation of 1 . 1 . Is their significant evidence to suggest that medication B has a larger HAM-D score decrease than medication A ? Test the hypotheses at the significance level α = 0 . 05 . Let μ 1 and μ 2 be the mean decrease in the HAM-D score for patients taking medication A, B (respectively). Example: Construct a 98% confidence interval for the difference in the mean decrease, and interpret.
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