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Understanding Confidence Intervals for the Mean with Known Standard Deviation

This guide explores the concept of confidence intervals for estimating population means when the standard deviation is known. It covers point estimates, properties of good estimators, and interval estimates. Key concepts such as unbiasedness, consistency, efficiency, and confidence levels (90%, 95%, and 99%) are discussed. Real-world examples demonstrate calculating confidence intervals and margins of error. The document also includes practice problems to enhance understanding of sample size and precision in estimates related to various scenarios.

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Understanding Confidence Intervals for the Mean with Known Standard Deviation

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  1. 7.1 confidence Intervals for the Mean When SD is Known

  2. -A point estimate is a specific numerical value that estimates a parameter. • -The best point estimate of the population mean µ is the sample mean X.

  3. 3 Properties of a good estimator • Unbiased: the mean of estimates is equal to the parameter being estimated. • Consistent: as the sample size increases, the value of the estimator approaches the value of the parameter. • Relatively Efficient: has the smallest variance

  4. Confidence Intervals • Interval Estimate: an interval (range) used to estimate the parameter. • -may or may not contain the parameter • Ex. 20 < µ < 25

  5. Confidence Intervals • Confidence Level- probability that the interval estimate will contain the parameter • -three common levels are 90%, 95%, and 99% • Confidence Interval- specific interval estimate of a parameter • -determined by using data and specific confidence levels • -as the percent increases, so does the range of the interval.

  6. Formula for the confidence interval X = sample mean For 90% confidence interval: N = sample size σ = standard deviation 95% confidence interval: 99% confidence interval:

  7. MARGIN of error • The maximum likely difference between the point estimate and the actual value of the parameter.

  8. A researcher wishes to estimate the number of days it takes an automobile dealer to sell a Chevrolet Aveo. A sample of 50 cars had a mean time on the dealer’s lot of 54 days. Assume the population standard deviation to be 6.0 days. Find the best point estimate of the population mean and the 95% confidence interval of the population mean.

  9. So, we can say with 95% confidence that the interval between 52.3 and 55.7 does contain the population mean.

  10. A survey of 30 emergency room patients found that the average waiting time for treatment was 174.3 minutes. • Assuming that the population standard deviation is 46.5 minutes, find the best point estimate of the population mean and the 99% confidence of the population mean.

  11. Practice problems: p. 366 • Numbers 11, 13, 14, 15, 17

  12. Determining Sample size E = margin of error N = sample size σ = standard deviation depends on confidence interval

  13. - A scientist wishes to estimate the average depth of a river. He wants to be 99% confident that the estimate is accurate within 2 feet. From a previous study, the standard deviation of the depths measured was 4.33 feet. How large a sample is required?

  14. Practice! • p. 367 numbers 21, 23, 24, 25

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