Class Notes 9 - Quiz A professor regularly gives multiple...

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Quiz: A professor regularly gives multiple choice quizzes with 5 questions. Over time, he has found the distribution of the number of wrong answers on his quizzes is as follows x P(x) 0 0.25 1 0.35 2 0.20 3 0.15 4 0.04 5 0.01 x p(x) x u p(x) 0 .25 1 .35 2 .20 3 .15 4 .04 5 .01 Variance of a discrete random variable x: 2 2 ( ) ( ) x x p x σ μ = - Standard deviation of x: 2 ( ) ( ) x x p x σ μ = - Quiz revisited: x p(x ) x×p(x) x-μ (x- μ) 2 (x- μ) 2 ×p(x) 0 .25 1 .35 2 .20 3 .15 4 .04 5 .01 Variance = Standard deviation = 6.4: How Likely Are the Possible Values of a Statistic? The Sampling Distribution Sample Proportions In statistical inference, we are usually more interested in the proportion of people who answer yes to a certain question, for example, rather than the count. We will use a sample proportion to estimate the true but unknown population proportion of successes in a binomial setting. Recall: X is the count of “successes” out of n independent binomial trials, each having probability p of success.
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Sample proportion of successes: number of successes ˆ number of observations x p n = = Be sure to distinguish between the proportion ˆ p and the count X. The proportion is a number between 0 and 1 while the count is a number between 0 and n. Note that this sample proportion is actually an average. In the numerator we have the number of successes, which is what you would get if you were to add up the string of 0’s and 1’s that represent failure and success, respectively. When you divide that by the number of observations, you effectively get an average. Sampling Distribution of ˆ p We want to know the distribution of all possible values the sample proportion ˆ p can take in repeated sampling. Normal approximation: The sampling distribution of ˆ p will be approximated by the normal distribution if both the expected number of successes and the expected number of failures are at least 15. That is, we need to check that np ≥ 15 and N(1-p) ≥ 15. Mean of the distribution of ˆ p : p Standard error of the distribution of ˆ p : (1 ) p p n - Note – we use the term standard error to refer to the standard deviation of a sampling distribution, and to distinguish it from the standard deviation of an ordinary probability distribution. Example: Suppose that 54% of the registered voters in the population support your favorite candidate – that is, if the election were today, your candidate will win. A survey before the election asks a random sample of 1000 registered voters whom they are planning to vote for the election. Find the probability
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