501_Lecture_05

# 501_Lecture_05 - Section 5.2 Sampling Distribution for...

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Section 5.2 Sampling Distribution for Counts and Proportions

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Preview Population distribution vs. sampling distribution Binomial distributions for sample counts Finding binomial probabilities: tables Binomial mean and standard variation Sample proportions Normal approximation for counts and proportions
Basic notions: A statistic from a random sample is a random variable (i.e. we will get a different value for each sample). Its distribution (when all samples are considered) is called the sampling distribution . The Population distribution of a variable is the distribution of its values among all the members of the population. This is often unknown. parameter: describes some feature of the population distribution

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Binomial distribution for sample counts Example: n = 2500 adults asked whether shopping is frustrating n is the number of trials X = 1650 answered Yes X is the number of “successes” p-hat = X/n = 0.66 is the sample proportion (of successes) Need to make sure we distinguish between the count and the sample proportion
Binomial Setting 1. Each observation falls in just two categories: Success/Failure Heads/Tails Yes/No 1. All observations are independent 2. Fixed number of trials, n 3. The probability of success, p , is the same in each trial The distribution of the (total) count of successes in this binomial setting is: Binomial distribution denoted B(n,p)

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Examples: Toss a fair coin 10 times and count the number X of heads Binomial or not? What about a biased coin? Deal 10 cards from a shuffled deck of 52. X is the number of spades. Binomial? Suggestions? Number of girls born among first 100 children in a (large) hospital this year Number of girls born in this hospital so far this year
Binomial distribution in statistical sampling SRS is not quite a Binomial setting Why? Check the 4 properties! However, if the population is 10 times larger than our sample n, then the number of successes in the sample is approximately Binomial. We say B(n,p) Here p is the population success rate usually unknown

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How to calculate Binomial probabilities? We will just use table C For given n and p, table gives the probability for k successes Table only gives p s of 0.5 or less If you have a p greater than 0.5, you need to switch the role of successes and failures.

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Binomial Example: Bill is the star player on his basketball team. Over his career, his free throw percentage is

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501_Lecture_05 - Section 5.2 Sampling Distribution for...

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