Lab 5a for Math 17 Coin Activity for CLT and Applications

Lab 5a for Math 17: Coin Activity for CLT and Applications Recall: The distribution formed by considering the value of a statistic for every possible sample of a given size n from the population is called the sampling distribution of the statistic. It is usually too difficult to enumerate all possible samples and compute all possible values of the statistic by hand, but we can approximate the distributions by taking a “large” number of samples (via simulation) to help visualize the distribution. Statistical theory helps us determine the distributions of some common sampling distributions. 1 Coin Activity Suppose we want to understand how the sample mean year on pennies behaves. The population of pennies we have available for investigation is a collection of 1002 pennies which were obtained from the UMass Five College Credit Union on August 25, 2010 ($10 in pennies was asked for). What do you think the distribution of year looks like for the population of pennies? Explain. Obtain a sample of 30 pennies, and compute the sample mean year for your sample of pennies. What value do you get? We note that due to time constraints, we are not sampling with replacement. (There is room here to record the years for your pennies.) Compare your mean value with the class (class graph). Are the values very different? What does the distribution of sample mean year look like based on the graph? Do you think looking at roughly 20 samples of size 30 is good enough to tell us about the distribution of sample mean year when n is 30? 1 Recall: The Sampling Distribution for p can be described as: approximately normal when np and n(1 − p) are both greater than or equal q to 10, with a mean denoted µp̂ = p, the population proportion, and a standard deviation σp̂ = p(1−p) n if a random sample was taken. And our new result from today: The Sampling Distribution for the sample mean, X̄ can be described as having a mean µX̄ = µ, the population mean, and a standard deviation σX̄ = √σn . The distribution is exactly normal if the parent population is normal. Finally, the Central Limit Theorem tells us the distribution will be approximately normal with the mean and standard deviation stated above if n is sufficiently large even if the population distribution is not normal. Again, having a random sample (randomization and independence condition) is necessary. 2 Try an Application Suppose 40 percent of the voters in a large city prefer candidate Q for mayor. A random sample of 2400 city voters is taken. a. What is the sampling distribution of the sample proportion of city voters who prefer candidate Q for mayor? Check that this distribution is valid. b. What is the probability that the sample taken results in a sample proportion of .426 or higher? c. If you obtained a sample proportion of .426, does that seem inconsistent with a belief that p=.4 as specified? (Your response should be based off your calculations in the first 2 parts.) 2