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Unformatted text preview: Biostatistics 100B Homework Solutions 1 January 17th, 2007 Solutions To Homework Assignment 1 General Comments: The solutions given below are (quite a bit) more extensive than would have been necessary to get full credit. I use the answer key as an opportunity to make important points, or mention commonly made mistakes. Nonetheless, the answer key should give you an idea of the type of solutions I would like to receive. Warmup Problems (1) Bad News Burgers: (a) This is one of those confusing situations where you could argue the hypotheses either way. I have asked you to give a reason why it might make sense for Hamburger Heavens null hypothesis to be that the meat is contaminated. Suppose I am the owner of the chain and it has just been announced in the news that people are getting sick at my restaurant. What will happen? I will lose lots of business UNLESS I CAN PROVE TO PROSPECTIVE CUSTOMERS THAT THE REMAINING MEAT IS SAFE. The thing I want to prove is my alternativenamely that the meat is not contaminated. My null hypothesis must be that the meat is contaminated. Another way of saying this is that I prefer to play it safe and not risk serving the meat until I am sure it is OKI assume the worst and try to prove the best. (Of course, if I were a consumer advocate, trying to shut the restaurant down I might need to prove there was the contamination, thereby reversing the hypotheses.) (b) A type I error occurs if you reject the null hypothesis when the null hypothesis is true. In this problem the null hypothesis is that the meat is contaminated. Therefore a type I error would consist of deciding that the hamburger is not contaminated when in fact it is. A type II error occurs if you fail to reject the null hypothesis when the null hypothesis is false. In this example you would make a type II error if you decided that the hamburger was contaminated when in fact it was not. The probability of a type I error is and the probability of a type II error is . In this problem it is much more serious to make a type I error than a type II error. If we make a type I error, contaminated burgers will be sold, many more people will get sick, and Hamburger Heaven will be in big trouble. If we make a type II error we will only lose a bit of money by failing to use some hamburger that was actually OK. Therefore we want to make as small as possible. (2) Casteneda vs. Partida: (a) The proportion of people in the population with Spanish names is .79 or 79%. If the juries are selected at random, they should be representative of the population, so 79% of jurors should have Spanish sounding names too. Thus p=.79. The sample proportion is p = 339 / 870 = . 39 (b) By the Central Limit Theorem, since n=870 is definitely large, (and in particular npq = (870)( . 79)( . 21) = 144 . 3 > 5) we know that under the null hypothesis p is approximately normally distributed with mean p and standard deviation radicalbig p (1 p ) /n . Thus we can standardize and use Z-scores to get the desired probability:....
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