Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 1

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 18 Sampling Distribution Models

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Modeling the Distribution of Sample Proportions A mean is not the only descriptive statistic whose sampling distribution is normal. The sampling distribution for proportions (or percentages) is also approximately normal. We would expect the histogram of the sample proportions to center at the true proportion, p, in the population.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Modeling the Distribution of Sample Proportions (cont.) It turns out that the histogram is unimodal, symmetric, and centered at p. More specifically, it’s an amazing and fortunate fact that a Normal model is just the right one for the histogram of sample proportions. To use a Normal model, we need to specify its mean and standard deviation. The mean of this particular Normal is at p.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Modeling the Distribution of Sample Proportions (cont.) When working with proportions, knowing the mean automatically gives us the standard deviation as well—the standard deviation we will use is So, the distribution of the sample proportions is modeled with a probability model that is

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Modeling the Distribution of Sample Proportions (cont.) A picture of what we just discussed is as follows:

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Assumptions and Conditions (cont.) 1.10% condition: If sampling has not been made with replacement, then the sample size, n, must be no larger than 10% of the population. 2.Success/failure condition: The sample size has to be big enough so that both and are greater than So, we need a large enough sample that is not too large.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Sampling Distribution Model for a Proportion (cont.) Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of is modeled by a Normal model with Mean: Standard deviation:

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Georgia State Troopers estimate that 75% of drivers on I-285 speed. They plan to set up a radar trap and check the speeds of 200 cars. About 95% of the time, what percentage of cars will be caught for speeding? A student rolls a pair of dice 200 times and keeps track of the number of sums of 10. What is the probability that she gets a sum of 10 at least 5% of the time? What is the probability that she gets a sum of 10 at least 15% of the time? A student rolls a pair of dice 100 times and keeps track of the number of sums of 7. What is the probability that she gets at least 20 sums of 7?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Tim makes 56% of his field goal attempts. If Tim shoots 39 times from the floor in tonight’s basketball game, what is the probability that he will make at least 25 shots? Erin flips a coin 100 times and observes 32 heads. Is this unusual? What is the smallest number of heads that would not be unusual? If Erin flips the coin 250 times, What is the largest number of heads that would not be unusual?