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Estimating a Population Proportion

Chapter 7 covers estimating parameters and determining sample sizes in statistics, focusing on confidence intervals for population proportions, means, and standard deviations. Key concepts include the margin of error, critical values, and conditions for the binomial distribution, emphasizing the process of calculating sample sizes for desired confidence levels. The chapter provides multiple examples demonstrating how to compute point estimates, confidence intervals, and interpretations of survey results.

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Elementary Statistics Chapter 7: Estimating Parameters and Determining Sample Sizes 7.1 Estimating a Population Proportion 1
Chapter 7: Estimating Parameters and Determining Sample Sizes 7.1 Estimating a Population Proportion 7.2 Estimating a Population Mean 7.3 Estimating a Population Standard Deviation or Variance 7.4 Bootstrapping: Using Technology for Estimates 2 Objectives: β€’ Find the confidence interval for a proportion. β€’ Determine the minimum sample size for finding a confidence interval for a proportion. β€’ Find the confidence interval for the mean when  is known. β€’ Determine the minimum sample size for finding a confidence interval for the mean. β€’ Find the confidence interval for the mean when  is unknown. β€’ Find a confidence interval for a variance and a standard deviation.
Margin of Error & Confidence Interval for Estimating a Population Proportion p & Determining Sample Size: When data from a simple random sample are used to estimate a population proportion p, the Margin of Error (Maximum Error of the Estimate) (Error Bound), denoted by E, is the maximum likely difference (with probability 1 – Ξ±, such as 0.95) between the observed (sample) proportion 𝑝 and the true value of the population proportion p. 3 𝑝 Β± 𝐸 𝑝 βˆ’ 𝐸 < 𝑝 < 𝑝 + 𝐸 𝑝 βˆ’ 𝑧𝛼 2 π‘π‘ž 𝑛 < 𝑝 < 𝑝 + 𝑧𝛼 2 π‘π‘ž 𝑛 7.1 Estimating a Population Proportion Summary πœŽπ‘ = π‘π‘ž 𝑛 β†’ 𝐸 = 𝑧𝛼 2 π‘π‘ž 𝑛 𝑛 = (𝑍𝛼/2)2π‘π‘ž 𝐸2 𝑛 = (𝑍𝛼/2)20.25 𝐸2 When no estimate of 𝒑 is known: 𝒑 = 𝒒 =0.5 Point estimate of p: 𝑝 = π‘ˆπΆπΏ+𝐿𝐢𝐿 2 , UCL: Upper Confidence Limit Margin of error: 𝐸 = π‘ˆπΆπΏβˆ’πΏπΆπΏ 2 , LCL: Lower Confidence Limi
Key Concept: Use a Sample proportion to make an inference about the value of the corresponding population proportion. A point estimate is a specific numerical value estimate of a parameter. (A point estimate is a single value (or point) used to approximate a population parameter. ) Properties of a good estimator: 1. The estimator should be an unbiased estimator. That is, the expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated. 2. The estimator should be consistent. For a consistent estimator, as sample size increases, the value of the estimator approaches the value of the parameter estimated. 3. The estimator should be a relatively efficient estimator; that is, of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance. β€’ The sample proportion 𝑝 is the best point estimate of the population proportion p. β€’ p = 𝑋 𝑁 = population proportion 𝑝 = π‘₯ 𝑛 (read p β€œhat”) = sample proportion, π‘ž = 1 βˆ’ 𝑝 x number of sample units that possess the characteristics of interest and n = sample size. X number of population units that possess the characteristics of interest and N = population size. β€’ Confidence Interval: We can use a sample proportion to construct a confidence interval estimate of the true value of a population proportion. β€’ Sample Size: We should know how to find the sample size necessary to estimate a population proportion. 7.1 Estimating a Population Proportion 4
A survey of 1500 adults indicated that 48% of them have a Facebook page. Based on that result, find the best point estimate of the proportion of all adults who have a Facebook page. Example 1 Solution Point estimate (PE) of p = 𝑝 = 0.48 Example 2: In a recent survey of 200 households, 60 had central air conditioning. Find 𝑝 and π‘ž , where 𝑝 is the proportion of households that have central air conditioning. x = 60 and n = 200 𝑝 = π‘₯ 𝑛 = 60 200 = 0.3 = 30% π‘ž = 1 βˆ’ 𝑝 = 1 βˆ’ 0.3 = 0.7 = 70% 5 𝑝 = π‘₯ 𝑛 , π‘ž = 1 βˆ’ 𝑝
Confidence Interval: A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI. This estimate may or may not contain the value of the parameter being estimated. Confidence Level: The confidence level is the probability 1 βˆ’ Ξ± (such as 0.95, or 95%) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times. (The confidence level is also called the degree of confidence, or the confidence coefficient.) The confidence level of 95% is the value used most often. 7.1 Estimating a Population Proportion Most Common Confidence Levels Corresponding Values of Ξ± 90% (or 0.90) confidence level: Ξ± = 0.10 95% (or 0.95) confidence level: Ξ± = 0.05 99% (or 0.99) confidence level: Ξ± = 0.01 6
Interpreting a Confidence Interval We must be careful to interpret confidence intervals correctly. There is a correct interpretation and many different and creative incorrect interpretations of the confidence interval 0.405 < p < 0.455. Correct: β€œWe are 95% confident that the interval from 0.405 to 0.455 actually does contain the true value of the population proportion p.” Wrong: β€œThere is a 95% chance that the true value of p will fall between 0.405 and 0.455.” Wrong: β€œ95% of sample proportions will fall between 0.405 and 0.455.” The Process Success Rate: A confidence level of 95% tells us that the process we are using should, in the long run, result in confidence interval limits that contain the true population proportion 95% of the time. 7.1 Estimating a Population Proportion Confidence Interval from 20 Different Samples 7 Confidence intervals can be used informally to compare different data sets, but the overlapping of confidence intervals should not be used for making formal and final conclusions about equality of proportions.
A critical value is the number on the borderline separating sample statistics that are significantly high or low from those that are not significant. The number zΞ±/2 is a critical value that is a z score with the property that it separates an area of Ξ±/2 in the right tail of the standard normal distribution. A standard z score can be used to distinguish between sample statistics that are likely to occur and those that are unlikely to occur. Such a z score is called a critical value. Critical values are based on the following observations: 8 7.1 Estimating a Population Proportion, Critical Values 1. Under certain conditions, the sampling distribution of sample proportions can be approximated by a normal distribution. 2. A z score associated with a sample proportion has a probability of Ξ±/2 of falling in the right tail. 3. The z score separating the right-tail region is commonly denoted by zΞ±/2 and is referred to as a critical value because it is on the borderline separating z scores from sample proportions that are likely to occur from those that are unlikely to occur.
Finding zΞ±/2 for a 95% Confidence Level Critical Values This is the most common critical value, and it is listed with two other common values in the table that follows.
Margin of Error & Confidence Interval for Estimating a Population Proportion p: When data from a simple random sample are used to estimate a population proportion p, the margin of error (also called the maximum error of the estimate ), denoted by E, is the maximum likely difference (with probability 1 – Ξ±, such as 0.95) between the observed (sample) proportion 𝑝 and the true value of the population proportion p. 𝑝 = sample proportion n = number of sample values zΞ±/2 = Critical value, z score separating an area of Ξ±/2 in the right tail of the standard normal distribution. Standard deviation of sample proportions: πœŽπ‘ = π‘π‘ž 𝑛 β†’ In practice, we don’t’ know the value of 𝒑, so we substitute 𝒑 β†’ πœŽπ‘ = π‘π‘ž 𝑛 E = margin of error 1. The sample is a simple random sample. 2. The conditions for the binomial distribution are satisfied: There is a fixed number of trials, the trials are independent, there are two categories of outcomes, and the probabilities remain constant for each trial. 3. There are at least 5 successes and at least 5 failures (np β‰₯ 5, and nq β‰₯ 5 ). 10 7.1 Estimating a Population Proportion 𝑝 Β± 𝐸 𝑝 βˆ’ 𝐸 < 𝑝 < 𝑝 + 𝐸 𝑝 βˆ’ 𝑧𝛼 2 π‘π‘ž 𝑛 < 𝑝 < 𝑝 + 𝑧𝛼 2 π‘π‘ž 𝑛 πœŽπ‘ = π‘π‘ž 𝑛 β†’ 𝐸 = 𝑧𝛼 2 π‘π‘ž 𝑛
Assume that a poll of 1487 adults showed that 43% of the respondents have Facebook pages. a. Find the margin of error E that corresponds to a 95% confidence level. b. Find the 95% confidence interval estimate of the population proportion p. c. Based on the results, can we safely conclude that fewer than 50% of adults have Facebook pages? Assuming that you are a newspaper reporter, write a brief statement that accurately describes the results and includes all of the relevant information. Example 2 Given: Binomial Distribution(BD) n = 1487, 𝑝 = 0.43 11 𝐸 = 𝑧𝛼 2 π‘π‘ž 𝑛 𝑏. 𝑝 Β± 𝐸 β†’ c. yes, because the interval of values is an interval that is completely below 0.50. π‘Ž. 𝐸 = 1.96 0.43(0.57) 1487 = 0.0252 Based on this poll of 1487 randomly selected adults in the United States, 43% of adults have Facebook pages. In theory, in 95% of such polls, the percentage should differ by no more than 2.52 percentage points in either direction from the percentage that would be found by interviewing all adults. 𝑝 Β± 𝐸 β†’ 𝑝 βˆ’ 𝐸 < 𝑝 < 𝑝 + 𝐸 TI Calculator: Confidence Interval: proportion 1. Stat 2. Tests 3. 1-prop ZINT 4. Enter: x, n & CL n𝑝 = (1487)(0.43) = 639 & nπ‘ž = (1487)(0.57) = 848 β‡Ύ n𝑝 β‰₯ 5 & nπ‘ž β‰₯ 5 0.4048 < p < 0.4552
A poll of 1007 randomly selected adults showed that 85% of respondents know what Twitter is. a. Find the margin of error E that corresponds to a 95% confidence level. b. Find the 95% confidence interval estimate of the population proportion p. c. Based on the results, can we safely conclude that more than 75% of adults know what Twitter is? d. Assuming that you are a newspaper reporter, write a brief statement that accurately describes the results and includes all of the relevant information. Example 3 Given: BD: n = 1007, 𝑝 = 0.85 12 π‘Ž. 𝐸 = 1.96 0.85 0.15 1007 = 0.0220545 𝑝 Β± 𝐸 β†’ b. 0.85 βˆ’ 0.0220545 < 𝑝 < 0.85 + 0.0220545 0.8279 < 𝑝 < 0.8721 c. Yes, because the interval of values is an interval that is completely above 0.75. d. 85% of U.S. adults know what Twitter is. That percentage is based on this poll of 1007 randomly selected adults. In theory, in 95% of such polls, the percentage should differ by no more than 2.21 percentage points in either direction from the percentage that would be found by interviewing all adults in the United States. 𝐸 = 𝑧𝛼 2 π‘π‘ž 𝑛 𝑝 Β± 𝐸 β†’ 𝑝 βˆ’ 𝐸 < 𝑝 < 𝑝 + 𝐸 n𝑝 = (1007)(0.85) = 855.951007 & nπ‘ž = (1007)(0.85) = 151.05 β‡Ύ n𝑝 β‰₯ 5 & nπ‘ž β‰₯ 5
Finding the Point Estimate and E from a Confidence Interval Point estimate of p: 𝑝 = π‘ˆπΆπΏ+𝐿𝐢𝐿 2 , UCL: Upper Confidence Limit Margin of error: 𝐸 = π‘ˆπΆπΏβˆ’πΏπΆπΏ 2 , LCL: Lower Confidence Limit 13 7.1 Estimating a Population Proportion Assume 70% of the 71 subjects that had smoke therapy did not smoke after 8 weeks and resulted in a 95% confidence interval [CI] of 58% to 81%. Find the point estimate p and the margin of error E. Example 4 Solution: 95% CI: 0.58 < p < 0.81 𝑝 = 0.81 + 0.58 2 = 0.695 𝐸 = 0.81 βˆ’ 0.58 2 = 0.115
Determining Sample Size: Finding the Sample Size Required to Estimate a Population Proportion: Objective: Suppose we want to collect sample data in order to estimate some population proportion. The question is how many sample items must be obtained? 14 7.1 Estimating a Population Proportion (solve for n by algebra) When no estimate of 𝒑 is not known: 𝒑 = 𝒒 =0.5 𝐸 = 𝑧𝛼 2 π‘π‘ž 𝑛 𝑛 = (𝑍𝛼/2)2 π‘π‘ž 𝐸2 𝑛 = (𝑍𝛼/2)2 0.25 𝐸2
Many companies are interested in knowing the percentage of adults who buy clothing online. How many adults must be surveyed in order to be 95% confident that the sample percentage is in error by no more than three percentage points? a. Use a recent result from the Census Bureau: 66% of adults buy clothing online. b. Assume that we have no prior information suggesting a possible value of the proportion. Example 5 Given: a. 𝑝 = 0.66, 𝐢𝐿 = 95%, 𝐸 = 0.03 π‘ž = 1 βˆ’ 𝑝 = 1 βˆ’ 0.66 = 0.34 15 𝐢𝐿 = 95% β†’ 𝑧𝛼 2 = 1.96 = (1.96)2(0.66)(0.34) (0.03)2 = 957.8 β†’ 𝑛 = 958 b. Assume 𝑝 = π‘ž = 0.5, 𝐢𝐿 = 95%, 𝐸 = 0.03 = (1.96)20.25 (0.03)2 = 1067.11 β†’ 𝑛 = 1068 𝑛 = (𝑍𝛼/2)2π‘π‘ž 𝐸2 , 𝑛 = (𝑍𝛼/2)20.25 𝐸2 𝑛 = (𝑍𝛼/2)2π‘π‘ž 𝐸2 𝑛 = (𝑍𝛼/2)20.25 𝐸2
A researcher wishes to estimate, with 95% confidence, the proportion of people who own a home computer. A previous study shows that 40% of those interviewed had a computer at home. The researcher wishes to be accurate within 2% of the true proportion. Find the minimum sample size necessary. Example 6 Given: a. 𝑝 = 0.4, 𝐢𝐿 = 95%, 𝐸 = 0.02 π‘ž = 1 βˆ’ 𝑝 = 1 βˆ’ 0.4 = 0.6 16 𝐢𝐿 = 95% β†’ 𝑧𝛼 2 = 1.96 = (1.96)2 (0.4)(0.6) (0.02)2 = 2304.96 β†’ 𝑛 = 2305 = 2304.96 𝑛 = (𝑍𝛼/2)2π‘π‘ž 𝐸2 , 𝑛 = (𝑍𝛼/2)20.25 𝐸2 𝑛 = (𝑍𝛼/2)2π‘π‘ž 𝐸2
A survey of 1404 respondents found that 323 students paid for their education by student loans. Find the 90% confidence of the true proportion of students who paid for their education by student loans. Given: Binomial Distribution(BD) n = 1404, π‘₯ = 323, 𝐢𝐿 = 90% 17 𝑝 = 323 1404 = 0.23 𝑝 Β± 𝐸 β†’ 0.23 βˆ’ 0.019 < 𝑝 < 0.23 + 0.019 0.211 < 𝑝 < 0.249 𝐸 = 1.645 0.23 0.77 1404 π‘ž = 1 βˆ’ 0.23 = 0.77 β‡Ύ n𝑝 β‰₯ 5 & nπ‘ž β‰₯ 5 You can be 90% confident that the percentage of students who pay for their college education by student loans is between 21.1% and 24.9%. Example 7 TI Calculator: Confidence Interval: proportion 1. Stat 2. Tests 3. 1-prop ZINT 4. Enter: x, n & CL = 0.019 𝐸 = 𝑧𝛼 2 π‘π‘ž 𝑛 𝑝 Β± 𝐸 β†’ 𝑝 βˆ’ 𝐸 < 𝑝 < 𝑝 + 𝐸
A survey of 1721 people found that 15.9% of individuals purchase religious books at a Christian bookstore. Find the 95% confidence interval of the true proportion of people who purchase their religious books at a Christian bookstore. 18 𝑝 βˆ’ 𝑧𝛼 2 π‘π‘ž 𝑛 < 𝑝 < 𝑝 + 𝑧𝛼 2 π‘π‘ž 𝑛 0.159 βˆ’ 1.96 0.159 0.841 1721 < 𝑝 < 0.159 + 1.96 0.159 0.841 1721 0.142 < 𝑝 < 0.176 Given: Binomial Distribution(BD) n = 1721, 𝑝 = 0.159, 𝐢𝐿 = 95% Example 8 𝐸 = 𝑍𝛼/2 π‘π‘ž 𝑛 𝑝 Β± 𝐸 β†’ 𝑝 βˆ’ 𝐸 < 𝑝 < 𝑝 + 𝐸