1 / 15

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

uriah-orr
Download Presentation

Lesson 9 - 1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related