Lecture_6 - Statistical Inference: Confidence Intervals...

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Statistical Inference: Confidence Intervals What are Point and Interval Estimates of Population Parameters?
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Learning Objectives 1. Point Estimate and Interval Estimate 2. Properties of Point Estimators 3. Confidence Intervals 4. Logic of Confidence Intervals 5. Margin of Error 6. Example
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Estimating with confidence Although the sample mean, , is a unique number for any particular sample, if you pick a different sample, you will probably get a different sample mean. In fact, you could get many different values for the sample mean, and virtually none of them would actually equal the true population mean, μ . x
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Learning Objective 1: Point Estimate and Interval Estimate A point estimate is a single number that is our “best guess” for the parameter An interval estimate is an interval of numbers within which the parameter value is believed to fall.
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Red dot: mean value of individual sample 95% of all sample means will be within roughly 2 standard deviations (2* σ /√ n ) of the population parameter μ. Because distances are symmetrical, this implies that the population parameter μ must be within roughly 2 standard deviations from the sample average , in 95% of all samples. n This reasoning is the essence of statistical inference. x
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Learning Objective 1: Point Estimate vs. Interval Estimate A point estimate doesn’t tell us how close the estimate is likely to be to the parameter An interval estimate is more useful It incorporates a margin of error which helps us to gauge the accuracy of the point estimate
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Confidence interval A level C confidence interval for a parameter has two parts: An interval calculated from the data, usually of the form estimate ± margin of error A confidence level C , which gives the probability that the interval will capture the true parameter value in repeated samples, or the success rate for the method.
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Learning Objective 2: Properties of Point Estimators Property 1: A good estimator has a sampling distribution that is centered at the parameter An estimator with this property is unbiased The sample mean is an unbiased estimator of the population mean The sample proportion is an unbiased estimator of the population proportion
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Learning Objective 2: Properties of Point Estimators Property 2: A good estimator has a small standard error compared to other estimators This means it tends to fall closer than other estimates to the parameter The sample mean has a smaller standard error than the sample median when estimating the population mean of a normal distribution
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Learning Objective 3: Confidence Interval A confidence interval is an interval containing the most believable values for a parameter The probability that this method produces an interval that contains the parameter is called the confidence level This is a number chosen to be close to 1, most commonly 0.95
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Learning Objective 4: Logic of Confidence Intervals To construct a confidence interval for a
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Lecture_6 - Statistical Inference: Confidence Intervals...

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