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Confidence Intervals

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Rate your confidence 0 - 100 Name my age within 10 years? within 5 years? within 1 year? Shooting a basketball at a wading pool, will make basket? Shooting the ball at a large trash can, will make basket? Shooting the ball at a carnival, will make basket?

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What happens to your confidence as the interval gets smaller? The larger your confidence, the wider the interval.

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Point Estimate singleUse a single statistic based on sample data to estimate a population parameter Simplest approach variationBut not always very precise due to variation in the sampling distribution

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Confidence intervals Are used to estimate the unknown population mean Formula: estimate + margin of error

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Margin of error Shows how accurate we believe our estimate is more preciseThe smaller the margin of error, the more precise our estimate of the true parameter Formula:

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Confidence level Is the success rate of the method used to construct the interval Using this method, ____% of the time the intervals constructed will contain the true population parameter

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What does it mean to be 95% confident? 95% chance that is contained in the confidence interval The probability that the interval contains is 95% The method used to construct the interval will produce intervals that contain 95% of the time.

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Found from the confidence level The upper z-score with probability p lying to its right under the standard normal curve Confidence leveltail areaz*.051.645.0251.96.0052.576 Critical value (z*).05 z*=1.645.025 z*=1.96.005 z*=2.576 90% 95% 99%

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Confidence interval for a population mean: estimate Critical value Standard deviation of the statistic Margin of error

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Steps for doing a confidence interval: 1)Assumptions – SRS from population Sampling distribution is normal (or approximately normal) Given (normal) Large sample size (approximately normal) Graph data (approximately normal) is known 2)Calculate the interval 3)Write a statement about the interval in the context of the problem.

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Statement: (memorize!!) We are ________% confident that the true mean context lies within the interval ______ and ______.

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Assumptions: Have an SRS of blood measurements Potassium level is normally distributed (given) known We are 90% confident that the true mean potassium level is between 3.01 and 3.39. A test for the level of potassium in the blood is not perfectly precise. Suppose that repeated measurements for the same person on different days vary normally with = 0.2. A random sample of three has a mean of 3.2. What is a 90% confidence interval for the mean potassium level?

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Assumptions: Have an SRS of blood measurements Potassium level is normally distributed (given) known We are 95% confident that the true mean potassium level is between 2.97 and 3.43. 95% confidence interval?

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99% confidence interval? Assumptions: Have an SRS of blood measurements Potassium level is normally distributed (given) known We are 99% confident that the true mean potassium level is between 2.90 and 3.50.

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What happens to the interval as the confidence level increases? the interval gets wider as the confidence level increases

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How can you make the margin of error smaller? z* smaller (lower confidence level) smaller (less variation in the population) n larger (to cut the margin of error in half, n must be 4 times as big) Really cannot change!

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A random sample of 50 SWH students was taken and their mean SAT score was 1250. (Assume = 105) What is a 95% confidence interval for the mean SAT scores of SWH students? We are 95% confident that the true mean SAT score for SWH students is between 1220.9 and 1279.1

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Suppose that we have this random sample of SAT scores: 950 1130 1260 1090 1310 1420 1190 What is a 95% confidence interval for the true mean SAT score? (Assume = 105) We are 95% confident that the true mean SAT score for SWH students is between 1115.1 and 1270.6.

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Find a sample size: If a certain margin of error is wanted, then to find the sample size necessary for that margin of error use: Always round up to the nearest person!

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The heights of SWH male students is normally distributed with = 2.5 inches. How large a sample is necessary to be accurate within +.75 inches with a 95% confidence interval? n = 43

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In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, takes calcium or placebo. The paper reports a mean seated systolic blood pressure of 114.9 with standard deviation of 9.3 for the placebo group. Assume systolic blood pressure is normally distributed. Can you find a z-interval for this problem? Why or why not?

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Some Cautions: The data MUST be a SRS from the population The formula is not correct for more complex sampling designs, i.e., stratified, etc. No way to correct for bias in data

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Cautions continued: Outliers can have a large effect on confidence interval Must know to do a z-interval – which is unrealistic in practice

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10.1 Estimating With Confidence

10.1 Estimating With Confidence

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