# 73 aside can you visualize it consider taking your

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73
Aside: Can you Visualize It? Consider taking your one random sample of size n and computing your one ^ p value. (As in the previous example, our one ^ p = 460/1000 = 0.46.) Suppose we did take another random sample of the same size, we would get another value of ^ p , say ___________. Now repeat that process over and over; taking one random sample after another; resulting in one ^ p value after another. Example picture showing the possible values when n = _______ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ^ p values Observations: How would things change if the sample size n were even larger, say n = _____? Suppose our first sample proportion turned out to be ^ p = _________ . Now imagine again repeating that process over and over; taking one random sample after another; resulting in many ^ p possible values. Example picture showing the possible values when n = _______ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ^ p values Observations: 74
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Let’s take a closer look at the sample proportion ^ p . The sample proportion is found by taking the number of “successes” in the sample and dividing by the sample size. So the count variable X of the number of successes is directly related to the proportion of successes as ^ p = X n . Earlier we studied the distribution of our first statistic, the count statistic X (the number of successes in n independent trials when the probability of a success was p ). We learned about its exact distribution called the Binomial Distribution . We also learned when the sample size n was large, the distribution of X could be approximated with a normal distribution. Normal Approximation to the Binomial Distribution If X is a binomial random variable based on n trials with success probability p , and n is large , then the random variable X is also approximately N ( np, np ( 1 p ) ) . Conditions : The approximation works well when both np and n (1 – p ) are at least 10. So any probability question about a sample proportion could be converted to a probability question about a sample count, and vice-versa. If n is small , we would need to convert the question to a count and use the binomial distribution to work it out. If n is large , we could convert the question to a count and use the normal approximation for a count, OR use a related normal approximation for a sample proportion (for large n ). The Stat 250 formula card summarizes this related normal approximation as follows: Let’s put this result to work in our next Try It! Problem. 76
Try It! Do Americans really vote when they say they do? To answer this question, a telephone poll was taken two days an election. From the 800 adults polled, 56% reported that they had voted. However, it was later reported in the press that, in fact, only 39% of American adults had voted. Suppose the 39% rate reported by the press is the correct population proportion. Also assume the responses of the 800 adults polled can be viewed as a random sample.