Sectio 7.2 Cofidece Iterval for a Proportio Before ay ifereces ca be made about a proportio, certai coditios must be satisfied: 1. The sample must be a SRS from the populatio of iterest. 2. The populatio must be at least 10 times the size of the sample. 3. The umber of successes must be 10, ad the umber of failures must be ( 1 10. The sample statistic for a populatio proportio is, so based o the formula for a CI, we have p ˆ margi of error. ---------------Width of the Cofidece Iterval------------------- -margi of error +margi of error Mea or Proportio How do we fid the margi of error if it is ot give to us? The margi of error is computed usig the critical value (a umber based o our level of cofidece) ad the stadard deviatio (stadard error) of the statistic. Critical Value: Whe the distributio is assumed to be ormal, our critical value is foud usig qorm i R. If it is ot ormal, we will use the t distributio (discussed later). The formula is: z 1 cofidece level qorm 2 Stadard Deviatio/Error: Whe workig with proportios, the stadard deviatio of the p ( 1 p) statistic is. Sice p is ukow, we will use the stadard error. To calculate the p ˆ(1 ) stadard error of, use the formula. So, p ˆ margi of error = z Sectio 7.2 Cofidece Iterval for a Proportio 1
Facts about Cofidece Itervals For smaller, the cofidece iterval becomes wider. For larger, the cofidece iterval becomes arrower. Icreasig the variace or icreasig the cofidece level will icrease the width of the cofidece iterval ad vice versa. Decreasig the variace or decreasig the cofidece level will decrease the cofidece iterval ad vice versa. Example 1: I the first eight games of this year s basketball seaso, Ley made 25 free throws i 40 attempts. a. What is ˆp, Ley s sample proportio of successes? Let s quickly check the coditios: 1. We have a SRS. Eve if it s ot stated i the problem, we ca assume it s a SRS. 2. The populatio is at least 10 times the sample (assume he wo t get hurt ad make may more free throws). 3. 40 25 / 40 25 10 ad ( 1 401 25 / 40 15 10 b. Fid ad iterpret the 90% cofidece iterval for Ley s proportio of free-throw success. 1 cofidece level z qorm 2 The z Cofidece Iterval: Iterpretatio: Sectio 7.2 Cofidece Iterval for a Proportio 2
Example 2: Mars Ic. claims that they produce M&Ms with the followig distributios: Brow 30% Red 20% Yellow 20% Orage 10% Gree 10% Blue 10% A bag of M&Ms was radomly selected from the grocery store shelf, ad the color couts were: Brow 22 Red 22 Yellow 22 Orage 12 Gree 15 Blue 15 Fid the 95% cofidece iterval for the proportio of yellow M&Ms i that bag. a. What is the proportio of yellow M&Ms i this bag? b. Fid ad iterpret the 95% cofidece iterval for the proportio of yellow M&Ms i this bag. 1 cofidece level z qorm 2 The z Cofidece Iterval: Sectio 7.2 Cofidece Iterval for a Proportio 3
Sometimes we are asked to fid the miimum sample size eeded to produce a particular margi of error give a certai cofidece level. Whe workig with a oe-sample proportio, we ca use the formula: Maximum ME z If is ukow, use a estimate of p. If p is ukow, just assume 50, 50, so use p = 0.5. Example 3: It is believed that 35% of all voters favor a particular cadidate. How large of a simple radom sample is required so that the margi of error of the estimate of the percetage of all voters i favor is o more tha 3% at the 95% cofidece level? 1 cofidece level p = Max ME = z qorm 2 Sectio 7.2 Cofidece Iterval for a Proportio 4
Example 4: A oil compay is iterested i estimatig the true proportio of female truck drivers based i five souther states. A statisticia hired by the oil compay must determie the sample size eeded i order to make the estimate accurate to withi 1% of the true proportio with 89% cofidece. What is the miimum umber of truck drivers that the statisticia should sample i these souther states i order to achieve the desired accuracy? p = ME = z 1 qorm cofidece level 2 ME / z 2 Sectio 7.2 Cofidece Iterval for a Proportio 5