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Confidence Intervals for a Mean when you have a “large” sample…

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The situation Want to estimate the actual population mean . But can only get , the sample mean. Find a range of values, L < < U, that we can be really confident contains . This range of values is called a “confidence interval.”

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Confidence Intervals for Proportions in Newspapers 18% of women, aged 18-24, think they are overweight. The “margin of error” is 5%. The “confidence interval” is 18% ± 5%. We can be really confident that between 13% and 23% of women, aged 18-24, think they are overweight.

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General Form of most Confidence Intervals Sample estimate ± margin of error Lower limit L = estimate - margin of error Upper limit U = estimate + margin of error Then, we’re confident that the population value is somewhere between L and U.

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Example Let X = number of high school friends Stat 250 students keep in touch with. True population mean = 5 friends. True population standard deviation = 5 friends. Take a random sample of 36 Stat 250 students. Calculate .

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Sampling Distribution of 55 + 2(0.83) 6.7 5 - 2(0.83) 3.3 0.95

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What does the sampling distribution tell us? 95% of the sample means will fall within 2 standard errors, or within 1.66 friends, of the true population mean = 5. Or, 95% of the time, the true population mean = 5 will fall within 2 standard errors, or within 1.66 friends, of the sample mean.

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Sampling Distribution of + 2( / n) - 2( / n) 0.95

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What does the sampling distribution tell us? 95% of the sample means will fall within 2 standard errors of the population mean Or, 95% of the time, the true population mean will fall within 2 standard errors of the sample mean. Use this last statement to create a formula.

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95% Confidence Interval for Formula in notation: Formula in English: Sample mean ± (2 × standard error of the mean)

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95% Confidence Interval for Formula in notation: Formula in English: Sample mean ± (2 × estimated standard error) 1. Formula OK as long as sample size is large (n 30) 2. Margin of error = 2 × standard error of the mean 3. 95% is called the “confidence level”

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Example A random sample of 32 students reported combing their hair an average of 1.6 times a day with a standard deviation of 1.3 times a day. In what range of values can we be 95% confident that , the actual mean, falls?

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What does 95% confident mean? + 2( / n) - 2( / n) 0.95 95% of all such confidence intervals will contain the true mean

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What if you want to be more (or less) confident? + Z( / n) - Z( / n) 0.98 1. Put confidence level in middle. 2. Subtract from 1. 3. Divide by 2 and put in tails. 4. Look up Z value. 0.01 Z 0.99 = 2.33

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Any % Confidence Interval for Formula in notation: Formula in English: Sample mean ± (Z × estimated standard error)

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Example A random sample of 64 students reported having an average of 2.4 roommates with a standard deviation of 4 roommates. In what range of values can we be 96% confident that , the actual mean, falls?

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Length of Confidence Interval Want confidence interval to be as narrow as possible. Length = Upper Limit - Lower Limit

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How length of CI is affected? As sample mean increases… As the standard deviation decreases… As we decrease the confidence level… As we increase sample size …

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Warning #1 Confidence intervals are only appropriate for random, representative samples. Problematic samples: –magazine surveys –dial-in surveys (1-900-vote-yes) –internet surveys (CNN QuickVote)

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Warning #2 The confidence interval formula we learned today is only appropriate for large samples (n 30). If you use today’s formula on a small sample, you’ll get a narrower interval than you should. Will learn correct formula for small samples.

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Warning #3 The confidence interval for the mean is a range of possible values for the population average. It says nothing about the range of individual measurements. The empirical rule tells us this.

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