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Confidence Intervals for a Mean

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

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Confidence Intervals for a Mean

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

  2. 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.”

  3. 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.

  4. General Form of most Confidence Intervals • Sample estimate ± margin of error • Lower limitL = 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.

  5. 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 .

  6. 5 - 2(0.83) 3.3 5 5 + 2(0.83) 6.7 Sampling Distribution of  0.95

  7. 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.

  8.  - 2(/n)   + 2(/n) Sampling Distribution of  0.95

  9. 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.

  10. 95% Confidence Interval for  Formula in notation: Formula in English: Sample mean ± (2 × standard error of the mean)

  11. 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”

  12. 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?

  13.  - 2(/n)   + 2(/n) What does 95% confident mean? 95% of all such confidence intervals will contain the true mean  0.95

  14.  - Z(/n)   + Z(/n) What if you want to be more (or less) confident? 1. Put confidence level in middle. 2. Subtract from 1. 3. Divide by 2 and put in tails. 4. Look up Z value. Z0.99 = 2.33 0.98 0.01 0.01

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

  16. 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?

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

  18. 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 …

  19. 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)

  20. 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.

  21. 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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