Section 6.4. Parameter and Statistic. How Likely Are the Possible Values of a Statistic? The Sampling Distribution

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1 Sectio 6.4 How Likely Are the Possible Values of a Statistic? The Samplig Distributio Parameter ad Statistic Parameter: A umerical summary of a populatio, such as a populatio proportio (p) or a populatio mea (µ). Statistic: A umerical summary of sample data, such as a sample proportio or a sample mea. Statistic estimates Parameter. Agresti/Frakli Statistics, e, of Agresti/Frakli Statistics, e, of Prior to coutig the votes, the proportio (p) i favor of recallig Goveror Gray Davis was a ukow parameter. A exit poll of 360 voters reported that the sample proportio i favor of a recall was That is x=706 voters i favor of a recall. The sample proportio=x/=706/360=0.54. If a differet radom sample of 360 voters were selected resultig 590 i favor or a (differet) sample proportio 675/360=0.53, which is differet from Imagie all the distict samples of 360 voters you could possibly get. The uder re-samplig sese, the sample proportio is a radom variable. Agresti/Frakli Statistics, e, 3 of Agresti/Frakli Statistics, e, 4 of Samplig Distributio Example: Samplig Distributio Questio: How do we kow that a sample statistic is a good estimate of a populatio parameter? Aswer: The samplig distributio. The samplig distributio of a statistic is the probability distributio that specifies probabilities for the possible values the statistic ca take. Which Brad of Pizza Do You Prefer? Two Choices: A or D. Assume that half of the populatio prefers A ad half prefers D. Parameter of iterest: p=populatio proportio. That is p=0.5. Take a radom sample of = 3 tasters. Agresti/Frakli Statistics, e, 5 of Agresti/Frakli Statistics, e, 6 of

2 Example: Samplig Distributio Example: Samplig Distributio Sample of size 3= (A,A,A) (A,A,D) (A,D,A) (D,A,A) (A,D,D) (D,A,D) (D,D,A) (D,D,D) No. Prefer Pizza A (x) 3 0 /3 /3 /3 Agresti/Frakli Statistics, e, 7 of Proportio (x/) 0 Sample Proportio 0 /3 Probability /8 3/8 3/8 /8 Agresti/Frakli Statistics, e, 8 of x x P( x) = C x p ( p) Example: Samplig Distributio Mea ad Stadard Deviatio of the Samplig Distributio of the Sample Proportio For a biomial radom variable with trials ad probability p of success for each, the samplig distributio of the proportio of successes has: Use biomial distributio: P( x) x x = C x p ( p) Mea = p ad stadard deviatio = p(- p) =stadard error To obtai these value, take the mea p ad stadard deviatio p ( p ) for the biomial distributio of the umber of successes ad divide by. Agresti/Frakli Statistics, e, 9 of Agresti/Frakli Statistics, e, 0 of Sample: Exit poll of 360 voters. =360 Suppose that exactly 50% of the populatio of all voters voted i favor of the recall. Describe the mea ad stadard deviatio of the samplig distributio of the umber i the sample who voted i favor of the recall. =360, p=0.50. µ = p = 360(0.50) = 580 σ = p( - p) = 360(0.50)(0.50) = 8. Agresti/Frakli Statistics, e, of Agresti/Frakli Statistics, e, of

3 Describe the mea ad stadard deviatio of the samplig distributio of the proportio i the sample who voted i favor of the recall. Mea = p = 0.50 Stadard Deviatio = p( p) = (0.50)(0.50) 360 = = If the populatio proportio supportig recall was 0.50, would it have bee ulikely to observe the exit-poll sample proportio of 0.54? Based o your aswer, would you be willig to predict that Davis would be recalled from office? Agresti/Frakli Statistics, e, 3 of Agresti/Frakli Statistics, e, 4 of Covert the sample proportio value of 0.54 to a z-score: ( ) z = = The sample proportio of 0.54 is more tha four stadard errors from the expected value of The sample proportio of 0.54 votig for recall would be very ulikely if the populatio support were p = Agresti/Frakli Statistics, e, 5 of A sample proportio of 0.54 would be eve more ulikely if the populatio support were less tha We have strog evidece that the actually p was large tha The exit poll gives strog evidece that Goveror Davis would be recalled. Agresti/Frakli Statistics, e, 6 of Fact: The samplig distributio of the sample proportio has a bell-shape with a mea µ = 0.50 ad a stadard deviatio σ = if p 5. (-p) 5. Recap: Summary of the Samplig Distributio of a Proportio (p) For a radom sample of size from a populatio with proportio p, the samplig distributio of the sample proportio has Mea = p ad stadard error = p(- p) If is sufficietly large such that the expected umbers of outcomes of the two types, p ad (- p), are both at least 5, the this samplig distributio has a bell-shape. Agresti/Frakli Statistics, e, 7 of Agresti/Frakli Statistics, e, 8 of 3

4 Sectio 6.5 How Close Are Sample Meas to Populatio Meas? The Samplig Distributio of the Sample Mea The sample mea, x, is a radom variable. The sample mea varies from sample to sample. By cotrast, the populatio mea, µ, is a sigle fixed umber. Agresti/Frakli Statistics, e, 9 of Agresti/Frakli Statistics, e, 0 of Mea ad Stadard Error of the Samplig Distributio of the Sample Mea For a radom sample of size from a populatio havig mea µ ad stadard deviatio σ, the samplig distributio of the sample mea has: Ceter described by the mea µ (the same as the mea of the populatio). Spread described by the stadard error, which equals the populatio stadard deviatio divided by the square root of the sample size: σ Example: How Much Do Mea Sales Vary From Week to Week? Daily sales at a pizza restaurat vary from day to day. The sales figures fluctuate aroud a mea µ = $900 with a stadard deviatio σ = $300. Agresti/Frakli Statistics, e, of Agresti/Frakli Statistics, e, of Example: How Much Do Mea Sales Vary From Week to Week? The mea sales for the seve days i a week are computed each week. The weekly meas are plotted over time. These weekly meas form a samplig distributio. Example: How Much Do Mea Sales Vary From Week to Week? What are the ceter ad spread of the samplig distributio? µ = $ σ = = 3 7 Agresti/Frakli Statistics, e, 3 of Agresti/Frakli Statistics, e, 4 of 4

5 Samplig Distributio vs. Populatio Distributio Stadard Error Kowig how to fid a stadard error gives us a mechaism for uderstadig how much variability to expect i sample statistics just by chace. Agresti/Frakli Statistics, e, 5 of Agresti/Frakli Statistics, e, 6 of Stadard Error The stadard error of the sample mea: σ As the sample size icreases, the deomiator icrease, so the stadard error decreases. With larger samples, the sample mea is more likely to fall close to the populatio mea. Cetral Limit Theorem Questio: How does the samplig distributio of the sample mea relate with respect to shape, ceter, ad spread to the probability distributio from which the samples were take? Agresti/Frakli Statistics, e, 7 of Agresti/Frakli Statistics, e, 8 of Cetral Limit Theorem For radom samplig with a large sample size, the samplig distributio of the sample mea is approximately a ormal distributio. This result applies o matter what the shape of the probability distributio from which the samples are take. Cetral Limit Theorem: How Large a Sample? The samplig distributio of the sample mea takes more of a bell shape as the radom sample size icreases. The more skewed the populatio distributio, the larger must be before the shape of the samplig distributio is close to ormal. I practice, the samplig distributio is usually close to ormal whe the sample size is at least about 30. Agresti/Frakli Statistics, e, 9 of Agresti/Frakli Statistics, e, 30 of 5

6 A Normal Populatio Distributio ad the Samplig Distributio If the populatio distributio is approximately ormal, the the samplig distributio is approximately ormal for all sample sizes. How Does the Cetral Limit Theorem Help Us Make Ifereces For large, the samplig distributio is approximately ormal eve if the populatio distributio is ot. This eables us to make ifereces about populatio meas regardless of the shape of the populatio distributio. Agresti/Frakli Statistics, e, 3 of Agresti/Frakli Statistics, e, 3 of 6