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Section 6.3: How to See the Future Goal: To understand how sample means vary in repeated samples. How do sample means vary? 141.8 109.2 30 Frequency Here’s a graph of the survival time of 72 guinea pigs, each injected with a drug. Note: The mean and std. dev. for the population are: 20 10 0 0 100 200 300 400 Survival time (days) 500 600 Understanding the POPULATION Data The survival times for the population of guinea pigs is skewed to the right. The mean, , is 109.2 The standard deviation, , is large! Distribution of Sample Means Suppose that lots and lots of us each randomly sampled 12 guinea pigs and found the average survival time for each set of 12 guinea pigs. We would each have a sample _ mean, x . Would all of our _ x values be the same? Use yesterday’s simulation data (consisting of 106 sample means) to see if there are any patterns. Graph of 106 sample means Count 20 10 0 100 150 sample means 200 Graph of the Sample Means What is the mean of the sample means, x ? _ What is the standard deviation of the sample means, _ ? x What is the shape of the graph of the sample means? Graph of the Sample Means What is the mean of the sample means, _ ? 142.01 x What is the standard deviation of the sample means, _ ? 27.94 x What is the shape of the graph of the sample means? Roughly bellshaped Comparing Population Data with Sample Mean Data What does this say in _ x words? _ What does this say in x words? The graph of the population data is skewed but the graph of the sample mean data is bell-shaped. Standard Error Since “standard deviation of the sample means” is a mouthful, we’ll instead call this quantity standard error. Remember, we have two standard deviations floating around – the first is the population standard deviation and the second is the standard error. The first describes how much spread there is in the population. The second describes how much spread there is in the sample means. Understanding Standard Error How does the population standard deviation relate to the standard error? _ x n What does this formula say? When the sample size is large, the standard error is small. When the sample size is small, the standard error is large. When n=1, the two values are equal. Why? Ex 2: Coin Problem Imagine you go home, collect all of the coins in your home, and make a graph of the age of each coin. This graph represents the graph of the population data. What do you expect its shape to be? Sample means of coins Take repeated samples, each of size 5 coins, and find the mean age of the coins. If you were to make a graph of the sample means, what would you expect it to look like? How about if instead you took samples of size 10? Or of size 25? Guinea Pigs and Coins In both situations the population graphs were severely skewed, yet the graph of the sample mean data was bell-shaped. In both cases the graphs of the sample mean data is centered at the population mean. In both cases the standard error is the population standard deviation divided by the square root of the sample size. Hmmmmm….. Coincidence? No, we couldn’t be that lucky! In fact, this is the Central Limit Theorem in action. Central Limit Theorem Suppose that a random sample of size n is taken from a large population in which the variable you are measuring has mean and standard deviation . Then, provided n is at least 30, the sampling distribution of the sample means is roughly bell-shaped, centered at the population mean, , with standard error equal to n . Since the graph of sample means will always be bell-shaped…. 68% of the sample means should come within one standard error of the center (population mean). 95% of the sample means should come within two standard errors of the center (population mean). 99.7% of the sample means should come within three standard errors of the center (population mean). Confidence Intervals to Estimate What is the average number of hours Ship students sleep per night during final exam week? Who is the population? What is the parameter we are interested in estimating? Confidence Intervals, Cont’d Estimate the population mean by taking a random sample of 50 college students and finding a sample mean. Suppose you find that the sample mean is 5.8 Confidence Intervals, cont’d. Can we say that is 5.8? If the sample was indeed random is it reasonable to believe is close to 5.8? Reason? Central Limit Theorem. So the 95% confidence interval to estimate is: _ x 2 n What is the formula for a 68% confidence interval? How about a 99.7% confidence interval?