Use the following information to answer the next five exercises: A poll of 1,200 voters asked what the most significant issue was in the upcoming election. Businesses that sell personal computers are interested in the proportion of households in the United States that own personal computers. Available online at http://www.pewinternet.org/~/media//Files/Questionnaire/2013/Methods%20and%20Questions_Teens%20and%20Social%20Media.pdf (accessed July 2, 2013). Let X = the number of people in the sample who have cell phones. Each possible sample gives us a different sample proportion and a different interval. The stated “± 3%” represents the maximum error bound. z 2. Suppose we wish to estimate the proportion of people with diabetes in a population or the proportion of people with hypertension or obesity. = Want to cite, share, or modify this book? These diagnoses are defined by specific levels of laboratory tests and measurements of blood pressure and body mass index, respectively. Next: Calculating the Sample Size n: Continuous and Binary Random Variables, Creative Commons Attribution 4.0 International License. Assume that the children in the selected class are a random sample of the population. 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. 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). We wish to construct a 90% confidence interval for the true proportion of California adults who feel that education and the schools is one of the top issues facing California. x A point estimate for the true population proportion is: A 90% confidence interval for the population proportion is _______. We are interested in the population proportion of voters who feel the economy is the most important. 0.025 If this survey were done by telephone, list three difficulties the companies might have in obtaining random results. the proportion of respondents who said they watched any television at all) What is p′ estimating? Available online at http://www.publicpolicypolling.com/Day2MusicPoll.pdf (accessed July 2, 2013). Comparing Two Independent Population Means, 50. Explain any differences between the values. Suppose 250 randomly selected people are surveyed to determine if they own a tablet. Unfortunately, there is no correction factor for cases where the sample size is small so np′ and nq' must always be greater than 5 to develop an interval estimate for p. Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. He surveys 500 accounts and finds that 300 are more than 30 days overdue. State the estimated distribution of X. X~________. Define random variables X and P′ in words. 2 ), x = the number of successes in the sample. Businesses that sell personal computers are interested in the proportion of households in the United States that own personal computers. ′ x = the number of successes in the sample = 421, p 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. z The lower limit is determined to be 0.08 and the upper limit is determined to be 0.16. A comparison between Confidence (Frequentist) versus Credible (Bayesian) Intervals for Proportions =0.842. This book is Creative Commons Attribution License Sixty-eight percent (68%) of students own an iPod and a smart phone. n We can say that there is a significant difference between the proportion of Asian adults who say that their families would welcome a white person into their families and the proportion of Asian adults who say that their families would welcome a black person into their families. If it were later determined that it was important to be more than 90% confident and a new survey were commissioned, how would it affect the minimum number you need to survey? OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. 2 We are interested in the population proportion of drivers who claim they always buckle up. To calculate the confidence interval, we must find p′, q′. Finally, we solve for Z. 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). Textbook content produced by OpenStax is licensed under a This Z-value can be found using a standard normal probability table. In the survey, 82% of manual laborers preferred trucks, 62% of non-manual wage earners preferred trucks, 54% of mid-level managers preferred trucks, and 26% of executives preferred trucks. Can we (with 75% confidence) conclude that at least half of all American adults believe this? In a sample of 300 students, 68% said they own an iPod and a smart phone. This Z-value can be found using a standard normal probability table. Contingency Tables and Probability Trees, 26. = =0.600 State the confidence interval, sketch the graph, and calculate the error bound. Compute a 90% confidence interval for the true percent of accounts receivable that are more than 30 days overdue, and interpret the confidence interval. , read "P hat". Explain what the ±2% means. This is because we do not know p, indeed, this is just what we are trying to estimate. During the first eight years of the study, 1.5% of the 451 members of the 50-Plus Fitness Association died. This is a handy trick to remember in finding Z-values for commonly used levels of confidence. ), p′ = the estimated proportion of successes or sample proportion of successes (p′ is a point estimate for p, the true population proportion, and thus q is the probability of a failure in any one trial. a. Compute a 90% confidence interval for the true percent of students who are against the new legislation, and interpret the confidence interval. We are interested in the proportion of people over 50 who ran and died in the same eight-year period. We estimate with 90% confidence that the true percent of all accounts receivable overdue 30 days is between 56.4% and 63.6%. Code to add this calci to your website 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. For example, a binomial distribution is the set of various possible outcomes and probabilities, for the number of heads observed when a coin is flipped ten times. Why? And then look up the probability for 1.51 standard deviations on the standard normal table. It is possible to create a confidence interval for the true population proportion following procedures similar to those used in creating confidence intervals for population means. 300 ( Investors in the stock market are interested in the true proportion of stocks that go up and down each week. z P α Available online at http://www.zogbyanalytics.com/news/299-americans-neither-worried-nor-prepared-in-case-of-a-disaster-sunyit-zogby-analytics-poll (accessed July 2, 2013). One of the questions asked was “What is the main problem facing the country?” Twenty percent answered “crime.” We are interested in the population proportion of adult Americans who feel that crime is the main problem. To interpret a confidence interval remember that the sample information is random - but there is a pattern to its behavior if we look at all possible samples. When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 90% confident that the population proportion is estimated to within 0.05? 2 Confidence Interval for a Proportion in One Sample. Finally, we solve for Z. The Dundee Dog Training School has a larger than average proportion of clients who compete in competitive professional events. We can say that there does not appear to be a significant difference between the proportion of Asian adults who say that their families would welcome a white person into their families and the proportion of Asian adults who say that their families would welcome a Latino person into their families. Compute a 97% confidence interval for the true percent of students who own an iPod and a smartphone. n They randomly surveyed 200 households and found that in 120 of them, the woman made the majority of the purchasing decisions. X is the number of “successes” where the woman makes the majority of the purchasing decisions for the household. 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. Alternate Wording: We estimate with 90% confidence that between 56.4% and 63.6% of ALL accounts are overdue 30 days. The Dundee Dog Training School has a larger than average proportion of clients who compete in competitive professional events. Binomial confidence interval calculation rely on the assumption of binomial distribution. =0.600, q 41 A Confidence Interval for A Population Proportion .

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