Hypothesis Testing: Binomial Experiment

AI Thread Summary
To determine if a cognitive behavioral (CB) program is more effective than a drug that cures 60% of depression cases, a binomial distribution approach is used with 15 trials. The null hypothesis posits that the cure rate is 60%, while the alternative suggests it is greater. The user attempts to find the minimum number of cured individuals needed to reject the null hypothesis at an alpha level of 0.10. By analyzing the binomial probabilities, they conclude that if 2 or fewer individuals are not cured, then at least 13 must be cured to support the claim of the CB program's effectiveness. The user seeks confirmation on their methodology and findings.
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Homework Statement



A drug company markets a medication that cures about 60% of cases with depression. A CB program is thought to be more effective. It was delivered to 15 depressed people. Determine the minimum number of cured people required to support the claim that the CB program is more effective than the drug. Use alpha=.10.

Homework Equations



nCx p^(x)q^(n-x)

The Attempt at a Solution



This is a binomial distribution problem.

- Upper tail test (H1: p>.6000)
n = number of trials = 15
p = probability of a success on a given trial = .6000
x = ?

I am trying to solve for x. However, I have no idea as to how to go about this. If I plug in the known values into the binomial distribution equation (written under "relevant equations") it becomes too difficult for me to solve, beyond the scope of the course I'm taking. I cannot use the normal approximation to solve the problem, because nxp does not equal 10.

Could someone please give me some detailed guidance? It would be greatly, greatly appreciated.
 
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I just had a thought. I could use the probability of failure instead of success (i.e. 1 - .6000 = .4000).

Then, I could look up n=15, p=.40 on the binomial distribution table. Starting at the lowest value of x (x=0, probability = .0005), it is evident that if one person was not cured, we could reject the null hypothesis because the p-value would be lower than alpha (.10).

I could work my way down the list, adding on each probability for the next highest value of x. When the probability exceeded the alpha level (this occurs at x=3), I would know I'd gone too high, because I could not reject the null hypothesis. The x value one down would be the key to my answer (x=2).

Thus, a maximum of 2 people must not be cured. To rephrase this in terms of the question, a minimum of 13 people must be cured.

Am I on the right track here?
 
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I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
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