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Lesson 9 - 1

Lesson 9 - 1. Logic in Constructing Confidence Intervals about a Population Mean where the Population Standard Deviation is Known. Objectives. Compute a point estimate of the population mean

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Lesson 9 - 1

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  1. Lesson 9 - 1 Logic in Constructing Confidence Intervals about a Population Mean where the Population Standard Deviation is Known

  2. Objectives • Compute a point estimate of the population mean • Construct and interpret a confidence interval about the population mean (assuming population σ is known) • Understand the role of margin of error in constructing a confidence interval • Determine the sample size necessary for estimating the population mean within a specified margin of error

  3. Vocabulary • Point Estimate – value of a statistic that estimates the value of a population parameter • Confidence Interval – for an unknown parameter is an interval of numbers (that the unknown falls between) • Level of Confidence – represents the expected proportion of intervals that will contain the parameter if a large number of samples is obtained. The level of confidence is denoted by (1- α) * 100% • α – represents the percentage the parameter falls outside the confidence interval (later known as a Type I error) • Robust – minor departures from normality do not seriously affect results • Z-interval – confidence interval

  4. Confidence Interval Estimates Point estimate (PE) ± margin of error (MOE) Point Estimate Sample Mean for Population Mean Sample Proportion for Population Proportion Expressed numerically as an interval [LB, UB]where LB = PE – MOE and UB = PE + MOE Graphically: PE MOE MOE _ x

  5. Margin of Error Factors • Level of confidence: as the level of confidence increases the margin of error also increases • Sample size: as the sample size increases the margin of error decreases (from Law of Large Numbers) • Population Standard Deviation: the more spread the population data, the wider the margin of error • MOE is in the form of measure of confidence • standard dev / sample size PE MOE MOE _ x

  6. σ E = zα/2 --- n Margin of Error, E The margin of error, E, in a (1 – α) * 100% confidence interval in which σ is known is given by where n is the sample size and zα/2 is the critical z-value. Note: The sample size must be large (n ≥ 30) or the population must be normally distributed.

  7. Using Standard Normal

  8. Measure of Confidence Z critical value: A value of the Z-statistic that corresponds to α/2 (1/2 because of two tails) for an α level of confidence PE MOE MOE _ x

  9. Interpretation of a Confidence Interval A (1-α) * 100% confidence interval indicates that , if we obtained many simple random samples of size n from the population whose mean, μ, is unknown, then approximately (1-α) * 100% of the intervals will contain μ. Note that is not a probability, but a level of the statistician’s confidence.

  10. Assumptions for Using Z CI Sample: simple random sample Sample Population: sample size must be large (n ≥ 30) or the population must be normally distributed. Dot plots, histograms, normality plots and box plots of sample data can be used as evidence if population is not given as normal Population σ: known (If this is not true on AP test you must use t-distribution!)

  11. A (1 – α) * 100% Confidence Interval about μ, σ Known Suppose a simple random sample of size n is taken from a population with an unknown mean μ and known standard deviation σ. A (1 – α) * 100% confidence interval for μ is given by σ Lower bound = x – zα/2 --- n σ Upper bound = x + zα/2 --- n where zα/2 is the critical z-value.

  12. Example 1 We have test 40 new hybrid SUVs that GM is resting its future on. GM told us the standard deviation was 6 and we found that they averaged 27 mpg highway. What would a 95% confidence interval about average miles per gallon be? PE ± MOE X-bar ± Z 1-α/2σ / √n 27 ± (1.96) (6) / √40 LB = 25.141 < μ < 28.859 = UB 95% confident that the true average mpg (μ) lies between LB and UB

  13. Example 2 GM told us the standard deviation for their new hybrid SUV was 6 and we wanted our margin of error in estimating its average mpg highway to be within 1 mpg. How big would our sample size need to be? (Z 1-α/2σ)² n = ------------- MOE² MOE = 1 n = (Z 1-α/2σ )² n = (1.96∙ 6 )² = 138.3 n = 139

  14. Summary and Homework • Summary • We can construct a confidence interval around a point estimator if we know the population standard deviation σ • The margin of error is calculated using σ, the sample size n, and the appropriate Z-value • We can also calculate the sample size needed to obtain a target margin of error • Homework • pg 458 – 465; 1-3, 10, 11, 19, 21, 23, 26, 37, 40, 47

  15. Homework • 1: sample size, confidence level, and standard deviation • 2: we widen our interval to become more confident that the true value is in there • 3: the greater the sample size the more the law of large numbers helps assure the point estimate is closer to the population parameter • 10: No, it does not look normal -- normality plot questionable • 11: Yes, normality plot ok and box plot symmetric-like • 19: σ= 13, x-bar=108, n=25 • 21 • 23 • 26: PE=103.4 minutes, c)(98.2,108.6) d) (99.0,107.8) e) decreased • 37: • 40: n=670 • 47: 4 times

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