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Statistics 101 Chapter 10 Section 10-1 We want to infer from the sample data some conclusion about a wider population that the sample represents. Inferential Statistics allows us this opportunity Statistical inference Provides methods for drawing conclusions about a population form sample data. We use probability to express strength of our conclusions Confidence Intervals and tests of significance Estimating with confidence SAT Math Scores in California Want to test population of 350,000 seniors SRS of 500 x = 461 What can you say about the mean score μ in the population of all 350,000 seniors? Results Law of large numbers Distribution close to normal Mean of sample = mean of population Standard deviation of x is σ / √500, let σ be 100 for known sd. Then, the standard deviation of x will be 4.5 Statistical Confidence The 68-95-99.7 rule says that in 95% of all samples, the mean score x for the sample will be within two standard deviations of the population mean. Whenever x is within + 9 points of the unknown μ, this happens 95% of all samples. What we can say We are 95% confident that the unknown mean SAT Math score for all California high school seniors lies between 452 and 470. How to find interval Interval = estimate + margin of error We use the symbol C to represent confidence interval. Question: Can you choose a different confidence interval instead of 95%. Exercises 10.2, 10.3 Confidence interval for a population mean The construction of a confidence interval for a population mean μ is appropriate when The data come from an SRS from the population of interest The sample distribution of x is approximately normal Finding z* To find 80% confidence interval, we catch the central 80%. We leave off 10% on both tails. So z* is point with area 0.1 to its right (and 0.9 to its left) under the normal curve. Search Table A to find the point with area 0.9 to its left. Critical values The number z* with probability p lying to its right under the standard normal curve is called the upper p critical value of the standard normal distribution Work through Example 10.5 Exercises 10.5 – 10.7 Confidence Interval Behavior High confidence says that our method almost always gives correct answers. Small margin of error says that we have pinned down the parameter quite precisely. Margin of error = z* σ / √ n Behavior z* gets smaller : to obtain a smaller margin of error from the same data, you must be willing to accept lower confidence. σ gets smaller: measures the variation in the population. Think of the variation among individuals in the population as noise that obscures the average value of μ. n gets larger: reduces margin of error. Since the n appears under the root sign, we must take four times as many observations in order to cut the margin of error in half. Exercises 10.8 – 10.11 Choosing the sample size. Example 10.7 on pages 551 and 552 Exercises 10.13 and 10.14