Ch7 Section 3: Confidence Intervals And Sample Size For Proportions
Ch7 Section 3: Confidence Intervals and Sample Size forProportionsBelief in Haunted PlacesA random sample of 205 collegestudents was asked if they believedthat places could be haunted, and65 of them responded yes. In thissection we’ll learn how to estimatethe true proportion of collegestudents who believe in thepossibility of haunted places.According to Time magazine, 37%of Americans believe that placescan be haunted. Source: Timemagazine, Oct. 2006.CH7: Confidence Intervals and Sample SizeSantorico - Page 242
TERMINOLOGY and DETAILS:The population proportion will be denoted by the letter p.The point estimate of the population proportion is the sampleproportion.ˆ , called p hat.We will symbolize the sample proportion by pFor large random samples, the central limit theorem tells usthat the sampling distribution of the sample proportion isapproximately normal. The distribution will have mean p and a standard deviation ofpq /n.CH7: Confidence Intervals and Sample Size Santorico - Page 243
The central limit theorem tells us what?!?If we take a bunch of samples theywill have proportions that fallaround the true populationproportion.AND, the sample proportions willhave a distribution that looksnormal over the samples.We can use the same ideas as in Section 7-1 to construct aconfidence interval for the population proportion p.CH7: Confidence Intervals and Sample SizeSantorico - Page 244
Let’s look at a simulation experiment to reinforce this concept: The main website for the simulation ishttp://socr.ucla.edu/htmls/SOCR Experiments.html You will want to select “Binomial Coin Experiment” in thedrop down menu on the right Here the coin represents the binomial event thatcorresponds to the proportion. You can change the sample size, n, and the population, p. And, create samples! Try different combinations and see what the samplingdistribution for the sample proportion looks like. This is the Central Limit Theorem!CH7: Confidence Intervals and Sample SizeSantorico - Page 245
Formula for a Confidence Interval for a ProportionAssumptions:1. The data are a random sample from the population.ˆ and nqˆ are each greater than or equal to 5.2. Both npFormula: pˆ z /2pˆ qˆnRounding Rule for a confidence interval for a proportion: round to 3decimal places. construct a confidence interval for a populationConcept: Once weproportion, what’s the probability our interval contains the truepopulation proportion?CH7: Confidence Intervals and Sample SizeSantorico - Page 246
Example: In a survey of 935 Denver residents, 60% said thatthey believed in aliens. Calculate a 95% confidence interval forthe proportion of Denver residents who believe in aliens.pˆ 0.6qˆ 0.4n 935z /2 1.96Check assumptions first! that the 935 represent a random sample 1. We must assumeof Denver residents.2. Since npˆ 935 0.6 561 and nqˆ 935 0.4 374 , wehave satisfied our second requirement that these be 5.CH7: Confidence Intervals and Sample SizeSantorico - Page 247
Next compute the interval:pˆ z /2ˆˆpq0.6 0.4 0.6 1.96n935 0.6 0.0314019 (0.569, 0.631)Finally, interpret: We are 95% confident that theproportion of Denver residents that believe in aliens isbetween 0.569 and 0.631.CH7: Confidence Intervals and Sample SizeSantorico - Page 248
Back to our motivating example: A random sample of 205college students was asked if they believed that places could behaunted, and 65 of them responded yes. Estimate the trueproportion of college students who believe in the possibility ofhaunted places with 99% confidence.pˆ qˆ n z /2 Check assumptions: CH7: Confidence Intervals and Sample Size Santorico - Page 249
Compute interval:pˆ z /2pˆ qˆnInterpret!CH7: Confidence Intervals and Sample SizeSantorico - Page 250
Sample Size for ProportionsSimilar to Section 7-1, we can determine the sample sizenecessary to achieve the desired precision of a confidenceinterval.Formula for Minimum Sample Size Needed for IntervalEstimate of a Population Proportion: z /2 2n pˆ qˆ , where E is the desired level of precision. If E necessary, round up to obtain a whole number.If an estimate of the proportion isn’t given, use pˆ 0.5 since thisis the worst case scenario (you will have to sample the mostsubjects to obtain the desired precision). CH7: Confidence Intervals and Sample SizeSantorico - Page 251
Example: How large a sample should be surveyed to estimatethe true proportion of college students who do laundry once aweek within 3.5% with 99% confidence? A previous studyplaced the proportion around 75%.222.575 z /2 ˆ ˆ n pq 0.75 0.25 1014.892 1015 0.035 E CH7: Confidence Intervals and Sample SizeSantorico - Page 252
Example: A research wishes to estimate the proportion ofexecutives who own a car phone. She wants to be 90%confident and be accurate within 5% of the true proportion.Find the minimum sample size necessary. z /2 2n pˆ qˆ E CH7: Confidence Intervals and Sample SizeSantorico - Page 253
that places could be haunted, and 65 of them responded yes. In this section we'll learn how to estimate the true proportion of college students who believe in the possibility of haunted places. According to Time magazine, 37% of Americans believe that places can be haunted. Source: Time magazine, Oct. 2006.
Jan 19, 2015 · Confidence Intervals with proportions a.k.a., “1-proportion z-intervals” AP Statistics Chapter 19. 1-proportion z-interval p . Microsoft PowerPoint - 19 Notes - Confidence Intervals wi
Section 7.2 – Confidence Interval for a Proportion 2 Facts about Confidence Intervals For smaller n, the confidence interval becomes wider. For larger n, the confidence interval becomes narrower. Increasing the variance or increasing the confidence level will increase the width of
90% or 95%). With known variance, formula for a confidence interval is: x z a all ZaJ2 Z all Z all 1.645 for 90% confidence intervals (significance level 10%, 5% in each tail) 1.960 for 95% confidence intervals (significance level 5%, 2.5% in each tail) 2.575 for 99% confidence intervals (significance level 1%, 0.5% in each tail)
Aug 16, 2018 · Biometrics & Biostatistics International Journal Research Article Open Access. Bootstrap confidence intervals for dissolution similarity factor f 2 398 Copy 2018 Citation: Islam MM, Begum M. Bootstrap confidence intervals for dissolution similarity factor f 2. Biom Biostat Int J. 2018;7(5):397‒403.
right end of each bar represent the 95 percent confidence intervals for each estimated population proportion. There is no difference between two career goals if their 95 percent confidence intervals overlap, even if the two bars differ in length. Other bar graphs in this report present the 95 percent confidence intervals in the same manner.
parameter. A confidence interval provides an estimate of the population parameter and the accompanying confidence level indicates the proportion of intervals that will cover the parameter. In other words, a confidence interval provides a range of values that would contain the true population par
Chapter 6: Estimation and Confidence Intervals. . the 95% confidence level interval,. Correct 19 times out of 20 1 time out of 20, by chance, our interval doesn’t contain the parameter (either below or above our interval) . confidence inte
0.1 0.0 x f(x) Sampling Distribution of the Mean 95% Confidence Interval: n 40 0.4 0.3 0.2 0.1 0.0 x f (x) Sampling Distribution of the Mean 95% Confidence Interval: n 20 When sampling from the same population, using a fixed confidence level, the larger the sample size, n, the narrower the confidence interval.
