l A matched pairs experiment compares the taste of instant with fresh-brewed cof

Question

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A matched pairs experiment compares the taste of instant with fresh-brewed coffee. Each subject tastes two unmarked cups of coffee, one of each type, in random order and states which he or she prefers. Of the 60 subjects who participate in the study, 21 prefer the instant coffee. Let p be the probability that a randomly chosen subject prefers fresh-brewed coffee to instant coffee. (In practical terms, p is the proportion of the population who prefer fresh-brewed coffee.)

(a)

Test the claim that a majority of people prefer the taste of fresh-brewed coffee. Report the large-sample z statistic. (Round your answer to two decimal places.)

Report its P-value. (Round your answer to four decimal places.)

Explanation / Answer

Solution:- Let X count the number of people out of 60 that select the fresh-brewed coffee. Now I want to show that most people (over half) prefer fresh-brewed coffee. Thus, we can not assume that most people prefer fresh-brewed coffee. We need a neutral assumption. A neutral assumption is that the preference is the same for instant versus fresh-brewed.

Let us assume that p = 0.5, that is half of the population n prefers fresh-brewed coffee over instant, evenly split. The measurement will be those people who said that they preferred fresh-brewed. We will try and see if there is any evidence that this proportion is more than 0.5, which would indicate that most people prefer fresh brewed coffee.

  H0: p = 0.5, Ha: p > 0.5.  pˆ = 39/60 = 0.65

Now we have 39 successes and we have 21 failures, both greater than 10 so we can use a normal approximation.

Test Statistic: Z = (0.65 - 0.5)/sqrt(0.5*0.5/60) = 2.32

P( pˆ > 0.65) = P(Z > 2.32) = 0.0102

The p-value is 0.0102

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