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The Distribution of Sample Means Section 9.3.1 Starter 9.3.1 • Write a sentence or two that answers each question: 1. What is a sample proportion? 2. What is a sampling distribution? 3. What is the significance of the mean of a sampling distribution of sample proportions? Fathom Demo • Fathom Demo 20 (The Distribution of Sample Proportions) shows the connection between: – Sample size and the spread of a sampling distribution – True population proportion and the center of a sampling distribution Today’s Objectives • Find the mean and standard deviation of the sampling distribution of x (sample means) for a given sample size n • Explain why averages are less variable than individual observations California Standard 16.0 Students know basic facts concerning the relation between the mean and the standard deviation of a sampling distribution and the mean and the standard deviation of the population distribution. Distribution of Sample Means • Suppose we draw an SRS of size n from a normally distributed population with mean µ and standard deviation σ. • Then the sampling distribution is: – Shape: approximately normal – Center: – Spread: X x n • Note: The population must be at least 10 times sample size to use this formula Example 9.7 • Suppose it is known that the heights of young women are N(64.5 in, 2.5 in). • Simulate measuring the heights of an SRS of 100 young women in your calculator. – randNorm(64.5, 2.5, 100) L1 • Report your results to me. Put class results in L2 – Run 1-Var Stats to find mean and s.d. of • What is the theoretical distribution of x x ? – The mean of the sampling distribution should be 64.5 – The standard deviation should be .25 Continue the problem • • • • If the sample mean you measured turned out to be 63.9 in, what is the probability of finding a group of 100 young women that short or shorter? normalcdf(0, 63.9, 64.5, .25) = .008 There was a 0.8% chance of getting these results if the population distribution was correct, so I find the results surprising. Possible reasons: 1. Perhaps sampling methodology was flawed. 2. Perhaps the original population distribution was wrong 3. Perhaps the 0.8% chance really happened! How do averages compare to individual observations? • Individual observations can vary widely. Consider the heights of Yao Ming and Verne Troyer (Mini-me). But when we look at several individuals and average the results, the high values tend to balance the low values. Thus averages don’t have such extremes: they are less variable than individual observations. • Now look again at the standard deviation formula: x n Today’s Objectives • Find the mean and standard deviation of the sampling distribution of x (sample means) for a given sample size n • Explain why averages are less variable than individual observations California Standard 16.0 Students know basic facts concerning the relation between the mean and the standard deviation of a sampling distribution and the mean and the standard deviation of the population distribution. Homework • Read pages 481 - 484 • Do problems 26 - 29