Things you need to know for 9.2 What is the mean and standard deviation for a binomial random variable? A binomial distribution is more or less mound-shaped and can be reasonably approximated by the normal distribution as long as….? Proportions are just another way of looking at count data. Ex: How many male students I have in AP Stats vs. the proportion of male students in class.

Sampling distribution of p-hat 9.2 Sample Proportions Sampling distribution of p-hat Allows us to find how good p-hat is an estimate of p We want to estimate the proportion of “successes” in the population, so take and SRS from the population of interest. Our estimator is p-hat = count of “successes” in sample (x) size of sample (n)

What proportion of U.S. teens know that 1492 was the year in which Columbus “discovered” America? A Gallup Poll found that 210 out of a random sample of 510 American teens aged 13 to 17 knew this historically important date. We use p-hat to gain information about the unknown population parameter p.

Derived from rules in Chapter 7  Because the mean of the sampling distribution of p-hat = p, p-hat is an unbiased estimator of p. Std. dev decreases as n increases.

ROT1: Used throughout the rest of the course whenever our interest is in drawing a sample to make inferences about a population. ROT2: Same conditions for using Normal approx. for binomial. Recall from 9.1: The sampling distribution of p-hat is approximately normal and is closer to normal when n is large. The accuracy of the Normal approximation improves as n increases.

Ex. 9.7 A polling organization asks an SRS of 1500 first-year college students whether they applied to other colleges. In fact, 35% of all first-year students applied to colleges besides the one they are attending. There are over 1.7 million first-year college students. What is the probability that the random sample of 1500 students will give a result within 2 percentage points of this true value?

Suppose you are going to roll a fair six-sided die 60 times and record p-hat, the proportion of times that a 1 or a 2 is showing. 1. Where should the distribution of the 60 p-hat values be centered? Justify your answer. 2. What is the standard deviation of the sampling distribution of p-hat, the proportion of all rolls of the die that show a 1 or a 2? 3. Describe the shape of the sampling distribution of p-hat. Justify your answer.  Power companies kill trees growing near their lines to avoid power failures due to falling limbs in storms. Applying a chemical to slow the growth of the trees is cheaper than trimming, but the chemical kills some of the trees. Suppose that one such chemical would kill 20% of sycamore trees. The power company tests the chemical on 250 sycamores. Consider these an SRS from the population of all sycamore trees. 4. What are the mean and standard deviation of the proportion of trees that are killed? 5. What is the probability that at least 60 trees (24% of the sample) are killed? (Remember to check that you can use the Normal approximation.)