Available online at http://www.rasmussenreports.com/public_content/lifestyle/sports/may_2013/52_say_big_time_college_athletics_corrupt_education_process (accessed July 2, 2013). If you decreased the allowable error bound, why would the minimum sample size increase (keeping the same level of confidence)? For example, a poll for a particular candidate running for president might show that the candidate has 40% of the vote within three percentage points (if the sample is large enough). A reporter is covering the release of this study for a local news station. The random variable P′ (read “P prime”) is the sample proportion, (Sometimes the random variable is denoted as , read “P hat”. ^ A comparison between Confidence (Frequentist) versus Credible (Bayesian) Intervals for Proportions. If it were later determined that it was important to be more than 95% confident and a new survey was commissioned, how would that affect the minimum number you would need to survey? Among Asians, 77% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person. Î± Since the requested confidence level is CL = 0.95, then α = 1 – CL = 1 – 0.95 = 0.05 = 0.025. If this survey were done by telephone, list three difficulties the companies might have in obtaining random results. During an election year, we see articles in the newspaper that state confidence intervals in terms of proportions or percentages. Businesses that sell personal computers are interested in the proportion of households in the United States that own personal computers. This Z-value can be found using a standard normal probability table. This gives rise to a proportion, meaning the percentage of the outcomes that are âsuccessesâ. Let X = the number of people in the sample who have cell phones. While the formulas are different, they are based upon the same mathematical foundation given to us by the Central Limit Theorem. A student polls his school to see if students in the school district are for or against the new legislation regarding school uniforms. Let p′ represent the sample proportion, x/n, where x represents the number of successes and n represents the sample size. If you are redistributing all or part of this book in a print format, The confidence interval for the true binomial population proportion is. The Central Limit Theorem for Sample Means, 36. = The possible outcomes are binary, either “success” or “failure”. x = the number of successes in the sample = 421. p′ = 0.842 is the sample proportion; this is the point estimate of the population proportion. = These diagnoses are defined by specific levels of laboratory tests and measurements of blood pressure and body mass index, respectively. The student’s t-table can also be used by entering the table at the 0.05 column and reading at the line for infinite degrees of freedom. Finally, we solve for Z. Calculating the Sample Size n: Continuous and Binary Random Variables, 45. = Z0.05 = 1.645. The t-distribution is the normal distribution at infinite degrees of freedom. The sampling error means that the true mean can be 2% above or below the sample mean. How do you know you are dealing with a proportion problem? Assuming all other variables are kept constant, as the confidence level increases, the area under the curve corresponding to the confidence level becomes larger, which creates a wider interval and thus a larger error. Suppose we want to lower the sampling error. Compute a 97% confidence interval for the true percent of students who own an iPod and a smartphone. = 0.05, Z Become a member and unlock all Study Answers Try it risk-free for 30 days x We are interested in the population proportion of adult Americans who are worried a lot about the quality of education in our schools. Refer to (Figure). Proportions are based upon the binomial probability distribution. = 0.025. It was shown that the binomial distribution could be fully understood if we knew only the probability of a success in any one trial, called p. The mean and the standard deviation of the binomial were found to be: It was also shown that the binomial could be estimated by the normal distribution if BOTH np AND nq were greater than 5. z Remembering the sampling distribution for the proportion from Chapter 7, the standard deviation was found to be: The confidence interval for a population proportion, therefore, becomes: Z(a2)Z(a2) is set according to our desired degree of confidence and pâ²(1âpâ²)npâ²(1âpâ²)n is the standard deviation of the sampling distribution. We have already seen that the sampling distribution of means is normally distributed. Use the following information to answer the next two exercises: Marketing companies are interested in knowing the population percent of women who make the majority of household purchasing decisions. We recommend using a Madden, Mary, Amanda Lenhart, Sandra Coresi, Urs Gasser, Maeve Duggan, Aaron Smith, and Meredith Beaton. It is possible that less than half of the population believe this. The lower limit is determined to be 0.08 and the upper limit is determined to be 0.16. We are interested in the population proportion of voters who feel the economy is the most important. Properties of Continuous Probability Density Functions, 32. 300 From the discussion above, it was found that the standardizing formula for the binomial distribution is: which is nothing more than a restatement of the general standardizing formula with appropriate substitutions for μ and σ from the binomial. = This project was supported by the National Center for Advancing Translational Sciences, National Institutes of Health, through UCSF-CTSI Grant Numbers UL1 … In a sample of 300 students, 68% said they own an iPod and a smart phone. Why? First, the underlying distribution has a binary random variable and therefore is a binomial distribution. Our mission is to improve educational access and learning for everyone. Interpreting Confidence Intervals. ‘z’ for 90% happens to be 1.64. The formula for the confidence interval for a population proportion follows the same format as that for an estimate of a population mean. =1-0.600=0.400, Since confidence level = 0.90, then Î± = 1 â confidence level = (1 â 0.90) = 0.10 â² The 5 P.M., Monday night, ages 8 to 12, beginning ice-skating class was picked. Interpretation of Regression Coefficients: Elasticity and Logarithmic Transformation, 73. The student's t-table can also be used by entering the table at the 0.05 column and reading at the line for infinite degrees of freedom. p(Z=1.51)=0.4345p(Z=1.51)=0.4345, p(Z)â 2=0.8690p(Z)â 2=0.8690 or 86.90%86.90%. The Field Poll. Suppose we wish to estimate the proportion of people with diabetes in a population or the proportion of people with hypertension or obesity. Confidence intervals can be calculated for the true proportion of stocks that go up or down each week and for the true proportion of households in the United States that own personal computers. ( 4.0 and you must attribute OpenStax. We can use the standard normal distribution, the reason Z is in the equation, because the normal distribution is the limiting distribution of the binomial. Of Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car. The variable p′ has a binomial distribution that can be approximated with the normal distribution shown here.