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Author(s): Brenda Gunderson, Ph.D., 2011 License: Unless otherwise noted, this material is made available under the terms of the Creative Commons Attribution–Non-commercial–Alike.

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Presentation on theme: "Author(s): Brenda Gunderson, Ph.D., 2011 License: Unless otherwise noted, this material is made available under the terms of the Creative Commons Attribution–Non-commercial–Alike."— Presentation transcript:

1 Author(s): Brenda Gunderson, Ph.D., 2011 License: Unless otherwise noted, this material is made available under the terms of the Creative Commons Attribution–Non-commercial–Alike 3.0 LiceShare nse: http://creativecommons.org/licenses/by-nc-sa/3.0/ We have reviewed this material in accordance with U.S. Copyright Law and have tried to maximize your ability to use, share, and adapt it. The citation key on the following slide provides information about how you may share and adapt this material. Copyright holders of content included in this material should contact open.michigan@umich.edu with any questions, corrections, or clarification regarding the use of content. For more information about how to cite these materials visit http://open.umich.edu/education/about/terms-of-use. Any medical information in this material is intended to inform and educate and is not a tool for self-diagnosis or a replacement for medical evaluation, advice, diagnosis or treatment by a healthcare professional. Please speak to your physician if you have questions about your medical condition. Viewer discretion is advised: Some medical content is graphic and may not be suitable for all viewers.

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3 More on the Standard Deviation of Standard deviation of : Interpret: approximately the average distance of the possible values (for repeated samples of same size n) from the true population proportion p. Problem = in practice we would not know the value of p.

4 The Standard Error of Standard error of : Interpret: Estimates, approximately, the average distance of the possible values (for repeated samples of same size n) from the true population proportion p. We can compute this! Use it to create a range of values …  (a few)s.e.( )

5 Example: Love at First Sight? pg 71 Random sample of n = 500 adults. 300 stated believe in love at first sight. a.Estimate the population proportion of adults that believe in love at first sight.

6 Example: Love at First Sight? Random sample of n = 500 adults. 300 stated believe in love at first sight. b.Next find the standard error of the estimate in part a:.

7 Example: Love at First Sight? Random sample of n = 500 adults. 300 stated believe in love at first sight. s.e.( ) = 0.022 b.Continued … Use the standard error provide an interval estimate for the population proportion p, with say, 95% confidence.

8 Chapter 10: Estimating Proportions with Confidence 10.2 An Overview of Confidence Intervals pg 73 Population & Population Parameter Sample & Sample statistic (estimate) Sample estimate will vary from one sample to the next. Standard error of sample estimate provides idea of how far away it would tend to vary from parameter value (on average).

9 General Form for a Confidence Interval General format for a confidence interval estimate is: Sample estimate ± (a few) standard errors  The “few” will depend on how confident we want to be.  “How confident” = confidence level.

10 Goal: Learn about a Population Proportion p Take random sample from population, estimate p with. Do you remember how values vary? Sampling Distribution of : If the sample size n is large enough, then the sample proportion is approximately …

11 Sampling Distribution of Follow along in your notes as we go through Steps 1 to 5.

12 Big Idea: Consider all possible r.s. of same large size n. Each possible random sample provides a possible sample proportion value. About 95% of possible sample proportion values will be in interval and for each of these values, the interval will contain the true proportion p. THUS…about 95% of the intervals will contain the true proportion p.

13 Initial 95% Confidence Interval for p: Dilemma: We don’t know the value of p. Solution: Replace p with, use called _________________________________. An approximate 95% confidence interval (CI) for population proportion p given by:

14 Try It! Getting Along with Parents pg 76 Gallup Youth Survey of n=501 randomly selected American teenagers asked about how well get along with parents. 54% of sample said “very well”. a.Sample proportion found to be 0.54. Give standard error for sample proportion and complete sentence. We would estimate the average distance between the possible _________________ values (from repeated samples) and _________________ to be about 0.022.

15 Try It! Getting Along with Parents Gallup Youth Survey of n=501 teenager: 54% of sample said “very well”. b.Compute a 95% CI for the population proportion of teenagers that get along very well with their parents.

16 Try It! Getting Along with Parents Gallup Youth Survey of n=501 teenager: 54% of sample said “very well”. The 95% CI for the population proportion of teenagers that get along very well with their parents is: 0.54 ± 0.44  (0.496, 0.584) c.Fill in: “Based on this sample, with 95% confidence, we would estimate that somewhere between ________ and ________ of all American teenagers think they get along very well with their parents.”

17 A = Yes or B = No Sample of n=501 teens  54% said “very well”. 95% CI for p is 0.54 ± 0.44  (0.496, 0.584) d.Can we say the probability that the above (already observed) interval ( 0.496, 0.584 ) contains the population proportion p is 0.95? that is,

18 A = Yes or B = No Sample of n=501 teens  54% said “very well”. 95% CI for p is 0.54 ± 0.44  (0.496, 0.584) e.Can we say that 95% of the time the population proportion p will be in the interval computed in part (b)?

19 What does the 95% Confidence Level Really Mean? The confidence level is used to describe the likeliness or chance that a yet-to-be constructed interval will actually contain the true population value in the following sense… Interpretation of the confidence level: If we consider all possible randomly selected samples of the same size from a population, the confidence level is the fraction or percent of those samples for which the confidence interval includes the population parameter.

20 Try It! Getting Along with Parents 95% CI for popul proportion of teenagers that get along very well with their parents. Based on r.s. n = 501 American teenagers. Interpreted the interval in part (c). Interpret the confidence level. The interval we found was computed with a method which if repeated over and over...

21 Try It! Completing a Graduate Degree page 78 Researcher has taken a r.s of n = 100 recent college graduates and recorded whether or not the student completed their degree in 5 years or less. Based on these data, a 95% confidence interval for the population proportion of all college students that complete their degree in 5 years or less is computed to be (0.62, 0.80).

22 a. How many of the 100 sampled college graduates completed degree in 5 yrs or less? A) 62 B) 71 C) 80 D) 95 E) Can’t be determined. 95% CI for p is (0.62, 0.80)

23 Valid interpretation of the 95% confidence level? i.There is about a 95% chance that the population proportion of students who have completed their degree in 5 years or less is between 0.62 and 0.80. ii.If sampling procedure were repeated many times, then approximately 95% of the resulting confidence intervals would contain the population proportion of students who have completed their degree in 5 years or less. iii.The probability that the population proportion p falls between 0.62 and 0.80 is 0.95 for repeated samples of the same size from the same population. 95% CI for p is (0.62, 0.80)

24 What about that Multiplier of 2? Exact multiplier for 95% confidence level is 1.96, was rounded to 2. Where does 1.96 come from? Use N(0,1) distrib. Other common levels are 90%, 98%, and 99%

25 Multiplier for 90% Confidence Level? Find the multiplier for 90% confidence level… The generic expression for the multiplier is z*.

26 Table 10.1 Multipliers pg 79 of lecture notes Page 481 of text Easiest to use Table A.2 with row = Infinite.. From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with permission.

27 Page 79 When the confidence level increases the value of the multiplier increases. So the width of the confidence interval also increases. To be more confident about the procedure, we need a wider interval.

28 Confidence Interval for a Population Proportion p page 79 where is sample proportion; z* is appropriate multiplier from N(0,1) distribution; and s.e.( ) = Conditions: 1.Sample is randomly selected sample from the popul. (Remember Fundamental Rule for Using Data for Inference). 2.Sample size n is large enough so the normal curve approx holds; i.e. np ≥ 10 and n(1–p) ≥ 10.

29 Try It! A 90% CI for p pg 79 Sample n = 501 Amer. teens: 54% get along very well with parents. Standard error was 2.2%. A 95% CI for population proportion of teens that get along very well: 49.6% to 58.4%. A 90% CI based on same sample: 50.4% to 57.6%. * narrower * still centered around 54%

30 The Conservative Approach page 80 Margin of error = z* s.e.( ) = z* is largest when = ½ = 0.5. Conservative CI for a population proportion p What happens when z*=2 for 95% confidence?

31 Choosing a Sample Size (Conservative) Margin of error = Solve for sample size: Here you do ALWAYS round UP to next integer!

32 Try It! Coke versus Pepsi pg 81 a.What is the margin of error for this interval?. Poll in Canada to estimate p, the proportion of all Canadian college students who prefer Coke over Pepsi. A 95% conservative confidence interval for p was found to be (0.62, 0.70).

33 Try It! Coke versus Pepsi b.What sample size would be necessary for a conservative 95% confidence interval for p with a margin of error of 0.03 (width of 0.06)? Think about (c) and click in answer

34 c. Poll repeated in U.S. (popul 10 times larger than Canada), but four times the number of people interviewed. Resulting 95% conservative CI will be: A) twice as wide B) 1/2 as wide C) 1/4 as wide D) 1/10 as wide E) same width


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