5/7/2014. Standard Error. The Sampling Distribution of the Sample Mean. Example: How Much Do Mean Sales Vary From Week to Week?

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1 Samplig Distributio Meas Lear. To aalyze how likely it is that sample results will be close to populatio values How probability provides the basis for makig statistical ifereces 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. 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: 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. 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? $ Stadard Error The stadard error of the sample mea: 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. 1

2 suppose x is a radom variable from a populatio with mea ad stadard deviatio. The for samples of size, mea of x-bar equals the populatio mea, or the stadard deviatio of x-bar equals the populatio stadard deviatio divided by the square root of sample size, or x if x is ormally distributed, so is x-bar regardless of sample size. if the sample size is large, x-bar is approximately ormally distributed, regardless of the distributio of x x Key fact i summary Thus, if either the variable uder cosideratio is ormally distributed, or the sample is large eough, the the distributio of all possible sample meas is at least close to a ormal distributio, with mea ad stadard deviatio Samplig distributio questios The Natioal Weather Service keeps records of sowfall i moutai rages. Records idicate that i a certai rage, the aual sowfall has a mea of 97 iches ad a stadard deviatio of 16 iches. Suppose the sowfalls are sampled durig radomly picked years. For samples of size 64 determie the mea ad stadard deviatio of. x Samplig distributio questios Suppose that replacemet times for washig machies are ormally distributed with a mea of 9.3 years ad a stadard deviatio of 1.1 years. Fid the probability that 70 radomly selected washig machies will have a mea replacemet time less tha 9.1 years. Samplig distributio questios For wome aged 18-24, systolic blood pressures (i mm Hg) are ormally distributed with a mea of ad a stadard deviatio of If 23 wome aged are radomly selected, fid the probability that their mea systolic blood pressure is betwee 119 ad 122. Samplig distributio questios Some experts estimate that the legth of time people live i a house has a mea of 7.2 years ad a stadard deviatio of 2.4 years. Explai why you could estimate the probability that 100 people selected at radom had lived i their houses for a average of 8 years or more, but you could ot estimate the probability that a idividual had doe so. 2

3 Cofidece Itervals Poit estimate ± margi of error The sample mea is the poit estimate of the populatio mea The exact stadard error of the sample mea is σ/ I practice, we estimate σ by the sample stadard deviatio, s How to Costruct a Cofidece Iterval for a Populatio Mea I practice, we do t kow the populatio stadard deviatio Substitutig the sample stadard deviatio s for σ to get se = s/ itroduces extra error To accout for this icreased error, we replace the z-score by a slightly larger score, the t-score Which method do I use for a CI for a Populatio Mea? If we have the populatio stadard deviatio, the we prefer to use z-iterval Otherwise, use t-iterval. The t-distributio is bell shaped ad symmetric about 0 The probabilities deped o the degrees of freedom, df The t-distributio has thicker tails ad is more spread out tha the stadard ormal distributio Example: ebay Auctios of Palm Hadheld Computers Do you ted to get a higher, or a lower, price if you give bidders the buy-it-ow optio? Cosider some data from sales of the Palm M515 PDA (persoal digital assistat) Durig the first week of May 2003, 25 of these hadheld computers were auctioed off, 7 of which had the buy-it-ow optio Example: ebay Auctios of Palm Hadheld Computers Buy-it-ow optio: Biddig oly: Sample Size for Estimatig a Populatio Mea I practice, sice you do t yet have the data, you do t kow the value of the sample stadard deviatio, s You must substitute a educated guess for s You ca use the sample stadard deviatio from a similar study z / 2 E 2 3

4 Weights of wome i oe age group are ormally distributed with a stadard deviatio σ of 14 lb. A researcher wishes to estimate the mea weight of all wome i this age group. Fid how large a sample must be draw i order to be 90 percet cofidet that the sample mea will ot differ from the populatio mea by more tha 3.9 lb. The weekly earigs of studets i oe age group are ormally distributed with a stadard deviatio of 36 dollars. A researcher wishes to estimate the mea weekly earigs of studets i this age group. Fid the sample size eeded to assure with 95 percet cofidece that the sample mea will ot differ from the populatio mea by more tha 2 dollars. What Are the Steps of a Sigificace Test about a Populatio Mea? Step 2: Hypotheses: The ull hypothesis has the form: H 0 : µ = µ 0 The alterative hypothesis has the form: H a : µ > µ 0 or H a : µ < µ 0 or H a : µ µ 0 Step 4: P-value The P-value summarizes the evidece It describes how uusual the data would be if H 0 were true Step 5: Coclusio We summarize the test by reportig ad iterpretig the P-value Example: Mea Weight Chage i Aorexic Girls A study compared differet psychological therapies for teeage girls sufferig from aorexia The variable of iterest was each girl s weight chage: weight at the ed of the study weight at the begiig of the study Oe of the therapies was cogitive therapy I this study, 29 girls received the therapeutic treatmet The weight chages for the 29 girls had a sample mea of 3.00 pouds ad stadard deviatio of 7.32 pouds Example: Mea Weight Chage i Aorexic Girls Example: Mea Weight Chage i Aorexic Girls How ca we frame this ivestigatio i the cotext of a sigificace test that ca detect a positive or egative effect of the therapy? Null hypothesis: o effect Alterative hypothesis: therapy has some effect The diet had a statistically sigificat positive effect o weight The effect, however, may be small i practical terms Aother example A public bus compay official claims that the mea waitig time for bus umber 14 durig peak hours is less tha 10 miutes. Kare took bus umber 14 durig peak hours o 18 differet occasios. Her mea waitig time was 7.5 miutes with a stadard deviatio of 2.2 miutes. At the 0.01 sigificace level, test the claim that the mea is less tha 10 miutes. 4

5 Example problem A light-bulb maufacturer advertises that the average life for its light bulbs is 900 hours. A radom sample of 15 of its light bulbs resulted i the followig lives i hours At the 10% sigificace level, do the data provide evidece that the mea life for the compay's light bulbs differs from the advertised mea? Examples A maufacturer claims that the mea weight of flour i its 32-ouce bags is 32.1 ouces. A z-test is performed to determie whether the mea weight is actually less tha this. The hypotheses are H0: μ = 32.1 ouces Ha: μ < 32.1 ouces. The mea weight for a sample of 45 bags of flour was 30.7 ouces. Suppose that the P-value correspodig to this sample data is Give a iterpretatio of the P-value. Would you feel cofidet i cocludig that the mea weight is less tha 32.1 ouces? I 1990, the average math SAT score for studets at oe school was 475. Five years later, a teacher wats to perform a hypothesis test to determie whether the average math SAT score of studets at the school has chaged. He picks a radom sample of 40 studets ad obtais their the mea math SAT score, which is 469. Do the data provide sufficiet evidece to coclude that the mea math SAT score for all studets at the school has chaged from the previous mea of 475? Perform the appropriate hypothesis test usig a sigificace level of =0.10. Assume that σ = 73. 5