construct a 90% confidence interval for the population mean

\(CL = 0.95 \alpha = 1 - 0.95 = 0.05 \frac{\alpha}{2} = 0.025 z_{\frac{\alpha}{2}} = 1.96.\) Use \(p = q = 0.5\). Available online at www.cdc.gov/growthcharts/2000thchart-us.pdf (accessed July 2, 2013). (5.87, 7.98) \(X\) is the number of unoccupied seats on a single flight. The confidence interval is (to three decimal places)(67.178, 68.822). A telephone poll of 1,000 adult Americans was reported in an issue of Time Magazine. As previously, assume that the population standard deviation is \(\sigma = 0.337\). Even though the three point estimates are different, do any of the confidence intervals overlap? From the problem, we know that \(\sigma = 15\) and \(EBM = 2\). That's a lot. When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 95% confident that the population proportion is estimated to within 0.03? Why? It means that should you repeat an experiment or survey over and over again, 95 percent of the time your results will match the results you get from a population (in other words, your statistics would be sound! This calculator will compute the 99%, 95%, and 90% confidence intervals for the mean of a normal population, given the sample mean, the sample size, and the sample standard deviation. If we know that the sample mean is \(68: EBM = 68.82 68 = 0.82\). It is denoted by n. Use the Student's t-distribution. Suppose that the firm decided that it needed to be at least 96% confident of the population mean length of time to within one hour. Ninety-five percent of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score. The \(z\)-score that has an area to the right of \(\dfrac{\alpha}{2}\) is denoted by \(z_{\dfrac{\alpha}{2}}\). In this survey, 86% of blacks said that they would welcome a white person into their families. For example, if we constructed 100 of these confidence intervals, we would expect 90 of them to contain the true population mean exam score. Suppose we know that a confidence interval is (42.12, 47.88). If many random samples were taken of size 14, what percent of the confidence intervals constructed should contain the population mean worth of coupons? Explain your choice. \(EBM = \left(z_{\dfrac{\alpha}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right)\). There is a known standard deviation of 7.0 hours. Suppose that our sample has a mean of \(\bar{x} = 10\), and we have constructed the 90% confidence interval (5, 15) where \(EBM = 5\). The standard deviation for this data to the nearest hundred is \(\sigma\) = $909,200. The 96% confidence interval is ($47,262, $456,447). When the sample size is large, s will be a good estimate of and you can use multiplier numbers from the normal curve. \(X =\) the number of people who feel that the president is doing an acceptable job; \(N\left(0.61, \sqrt{\frac{(0.61)(0.39)}{1200}}\right)\). So, to capture this uncertainty we can create a confidence interval that contains a range of values that are likely to contain the true mean weight of the turtles in the population. You want to estimate the mean height of students at your college or university to within one inch with 93% confidence. Arrow down to Calculate and press ENTER. It concluded with 95% confidence that 49% to 55% of Americans believe that big-time college sports programs corrupt the process of higher education. Refer to Exercise. Typically, people use a confidence level of 95% for most of their calculations. It randomly surveys 100 people. That is, theres only a 5% chance that the true population mean weight of turtles is greater than 307.25 pounds or less than 292.75 pounds. OR, from the upper value for the interval, subtract the lower value. Among Asians, 77% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person. \(CL = 0.95 \alpha = 1 - 0.95 = 0.05 z_{\frac{\alpha}{2}} = 1.96\). (The area to the right of this \(z\) is 0.125, so the area to the left is \(1 0.125 = 0.875\).). Construct a 90% confidence interval for the population proportion of people who feel the president is doing an acceptable job. The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is \(\pm 3%\). Assume that the numerical population of GPAs from which the sample is taken has a normal distribution. Metadata Description of Candidate Summary File. U.S. Federal Election Commission. Construct a 92% confidence interval for the population mean number of unoccupied seats per flight. For a two-tailed 95% confidence interval, the alpha value is 0.025, and the corresponding critical value is 1.96. Sample Variance In one to three complete sentences, explain what the 3% represents. Is the mean within the interval you calculated in part a? Construct confidence interval for P1 Pz at the given level of coniidence X1 = 25,n1 = 225,X2 = 38, 12 305, 90% confidence The researchers are 90% confident the difference between the two population proportions Pz, is between (Use ascending order: Type an integer or decimal rounded t0 three decimal places as needed ) and Assume the population has a normal distribution. To get a 90% confidence interval, we must include the central 90% of the probability of the normal distribution. ), \(EBM = (1.96)\left(\dfrac{3}{\sqrt{36}}\right) = 0.98\). Remember, in this section we know the population standard deviation . Find the 95% Confidence Interval for the true population mean for the amount of soda served. These intervals are different for several reasons: they were calculated from different samples, the samples were different sizes, and the intervals were calculated for different levels of confidence. Construct a 95% confidence interval for the population mean worth of coupons. Yes this interval does not fall less than 0.50 so we can conclude that at least half of all American adults believe that major sports programs corrupt education but we do so with only 75% confidence. Construct a 95% confidence interval for the population proportion who claim they always buckle up. In a recent Zogby International Poll, nine of 48 respondents rated the likelihood of a terrorist attack in their community as likely or very likely. Use the plus four method to create a 97% confidence interval for the proportion of American adults who believe that a terrorist attack in their community is likely or very likely. An example of how to calculate a confidence interval for a mean. A sample of 15 randomly selected students has a grade point average of 2.86 with a standard deviation of 0.78. Leave everything the same except the sample size. Without performing any calculations, describe how the confidence interval would change if the confidence level changed from 99% to 90%. Construct a 95% confidence interval for the population mean length of engineering conferences. Short Answer. The reporter claimed that the poll's " margin of error " was 3%. A Leadership PAC is a PAC formed by a federal politician (senator or representative) to raise money to help other candidates campaigns. The sample size is less than 30. They randomly surveyed 400 drivers and found that 320 claimed they always buckle up. Construct a 90% confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets. If we want to construct a confidence interval to be used for testing the claim, what confidence level should be used for the confidence . (This is the value of \(z\) for which the area under the density curve to the right of \(z\) is 0.035. If we decrease the sample size \(n\) to 25, we increase the error bound. 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? To find the 98% confidence interval, find \(\bar{x} \pm EBM\). Decreasing the sample size causes the error bound to increase, making the confidence interval wider. The population standard deviation is six minutes and the sample mean deliver time is 36 minutes. If the confidence level (\(CL\)) is 95%, then we say that, "We estimate with 95% confidence that the true value of the population mean is between 4.5 and 9.5.". The population standard deviation for the height of high school basketball players is three inches. Which? Find a confidence interval estimate for the population mean exam score (the mean score on all exams). (2.41, 3.42) (2.37, 3.56) (2.51, 3.21) (2.28, This problem has been solved! The sample mean is 23.6 hours. The Federal Election Commission collects information about campaign contributions and disbursements for candidates and political committees each election cycle. Construct a 95% confidence interval for the population mean height of male Swedes. Decreasing the confidence level decreases the error bound, making the confidence interval narrower. \[CL + \dfrac{\alpha}{2} + \dfrac{\alpha}{2} = CL + \alpha = 1.\nonumber \], The interpretation should clearly state the confidence level (\(CL\)), explain what population parameter is being estimated (here, a population mean), and state the confidence interval (both endpoints). \(N 7.9\left(\frac{2.5}{\sqrt{20}}\right)\). How to interpret a confidence interval for a mean. When \(n = 25: EBM = \left(z_{\dfrac{\alpha}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right) = (1.645)\left(\dfrac{3}{\sqrt{25}}\right) = 0.987\). However, it is more accurate to state that the confidence level is the percent of confidence intervals that contain the true population parameter when repeated samples are taken. A. What is the error bound? In words, define the random variable \(X\). Why or why not? The Specific Absorption Rate (SAR) for a cell phone measures the amount of radio frequency (RF) energy absorbed by the users body when using the handset. x = 39.9, n = 45, s = 18.2, 90% confidence E = Round to two decimal places if necessary <? Use the point estimate from part a and \(n = 1,000\) to calculate a 75% confidence interval for the proportion of American adults that believe that major college sports programs corrupt higher education. Suppose that a 90% confidence interval states that the population mean is greater than 100 and less than 200. For example, when \(CL = 0.95, \alpha = 0.05\) and \(\dfrac{\alpha}{2} = 0.025\); we write \(z_{\dfrac{\alpha}{2}} = z_{0.025}\). The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The confidence level, \(CL\), is the area in the middle of the standard normal distribution. Remember, in this section we already know the population standard deviation . Find a 98% confidence interval for the true (population) mean of the Specific Absorption Rates (SARs) for cell phones. The confidence level would increase as a result of a larger interval. A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes. A confidence interval for a mean gives us a range of plausible values for the population mean. the effective length of time for a tranquilizer, the mean effective length of time of tranquilizers from a sample of nine patients. Assume the underlying population is normally distributed. The main task for candidates lies in their ability to construct and interpret a confidence interval. If we took repeated samples, approximately 90% of the samples would produce the same confidence interval. Create a confidence interval for the results of this study. The life span of the English Bulldog is approximately Normal with a mean of 10.7 years. Find a 90% confidence interval estimate for the population mean delivery time. This means \(\bar{X}\) is normally distributed, that is, \(\bar{X} \sim N(\mu_{x},\dfrac{\sigma}{\sqrt{n}})\). Create a 95% confidence interval for the mean total individual contributions. Assume the underlying distribution is approximately normal. Construct a 95% confidence interval for the population mean time to complete the tax forms. The 98% confidence interval of the population mean amount of mercury in tuna sushi is equal to (0.287 ppm, 1.151 ppm) . \(CL = 0.95\) so \(\alpha = 1 CL = 1 0.95 = 0.05\), \(\dfrac{\alpha}{2} = 0.025 z_{\dfrac{\alpha}{2}} = z_{0.025}\). Assume the underlying population is normal. The following table shows the z-value that corresponds to popular confidence level choices: Notice that higher confidence levels correspond to larger z-values, which leads to wider confidence intervals. A 90% confidence interval for a population mean is determined to be 800 to 900. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Explain in a complete sentence what the confidence interval means. If the firm wished to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make? If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean. If we don't know the sample mean: \(EBM = \dfrac{(68.8267.18)}{2} = 0.82\). We estimate with 95% confidence that the true population mean for all statistics exam scores is between 67.02 and 68.98. Statology Study is the ultimate online statistics study guide that helps you study and practice all of the core concepts taught in any elementary statistics course and makes your life so much easier as a student. Increasing the confidence level increases the error bound, making the confidence interval wider. (b) Construct the 90% confidence interval for the population mean if the sample size, n, is 25. Statistics Statistical Inference Overview Confidence Intervals 1 Answer VSH Feb 22, 2018 Answer link A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Why? The 90% confidence interval is (67.18, 68.82). The population standard deviation is known to be 0.1 ounce. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. The sample mean is seven, and the error bound for the mean is 2.5: \(\bar{x} = 7\) and \(EBM = 2.5\), The confidence interval is (7 2.5, 7 + 2.5) and calculating the values gives (4.5, 9.5). Suppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of six minutes. The range can be written as an actual value or a percentage. Use the Student's \(t\)-distribution. The error bound and confidence interval will decrease. Confidence Interval Calculator for the Population Mean. Arrow down and enter the name of the list where the data is stored. We estimate with 98% confidence that the true SAR mean for the population of cell phones in the United States is between 0.8809 and 1.1671 watts per kilogram. Assume the underlying population is normal. You can choose the method that is easier to use with the information you know. What value of 2* should be used to construct a 95% confidence interval of a population mean? The 90% confidence interval is (67.1775, 68.8225). Arrow down and enter the following values: The confidence interval is (to three decimal places) (0.881, 1.167). (Notice this is larger than the z *-value, which would be 1.96 for the same confidence interval.) It is possible that less than half of the population believe this. Confidence intervals are an important reminder of the limitations of the estimates. Construct a 90% confidence interval for the mean GPA of all students at the university. The mean length of the conferences was 3.94 days, with a standard deviation of 1.28 days. You need to find \(z_{0.01}\) having the property that the area under the normal density curve to the right of \(z_{0.01}\) is \(0.01\) and the area to the left is 0.99. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study. ). The sample size would need to be increased since the critical value increases as the confidence level increases. You want to estimate the true proportion of college students on your campus who voted in the 2012 presidential election with 95% confidence and a margin of error no greater than five percent. What is 90% in confidence interval? The concept of the confidence interval is very important in statistics ( hypothesis testing) since it is used as a measure of uncertainty. The sample mean, x \bar{x} x , is determined to be 104.3 and the sample standard deviation, s, is determined to be 15.9. Suppose that the insurance companies did do a survey. Use the following information to answer the next three exercises: According to a Field Poll, 79% of California adults (actual results are 400 out of 506 surveyed) feel that education and our schools is one of the top issues facing California. We estimate with 93% confidence that the true SAR mean for the population of cell phones in the United States is between 0.8035 and 1.0765 watts per kilogram. To find the confidence interval, you need the sample mean, \(\bar{x}\), and the \(EBM\). The Federal Election Commission (FEC) collects information about campaign contributions and disbursements for candidates and political committees each election cycle. Sample mean (x): Sample size: Compare the error bound in part d to the margin of error reported by Gallup. Forty-eight male Swedes are surveyed. A sample of 15 randomly selected math majors has a grade poi Algebra: Probability and statistics Solvers Lessons Answers archive Click here to see ALL problems on Probability-and-statistics This is incorrect. This page titled 8.E: Confidence Intervals (Exercises) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. Because you are creating a 98% confidence interval, \(CL = 0.98\). State the confidence interval. Answer: (4.68, 4.92) The formula for the confidence interval for one population mean, using the t- distribution, is In this case, the sample mean, is 4.8; the sample standard deviation, s, is 0.4; the sample size, n, is 30; and the degrees of freedom, n - 1, is 29. Construct a 95% confidence interval for the population proportion of adult Americans who are worried a lot about the quality of education in our schools. How would you interpret this statement? Which distribution should you use for this problem? x=60 =15 n=20 N=200 The 90% Calculus and Above Ask an Expert Answers to Homework Calculus Questions Answered in 5 minutes by: Ask Your Own Calculus and Above Question Kofi Ask Your Own Calculus and Above Question Ask Your Own Calculus and Above Question Why? Since we increase the confidence level, we need to increase either our error bound or the sample size. If we want to be 95% confident that the sample mean age is within two years of the true population mean age of Foothill College students, how many randomly selected Foothill College students must be surveyed? Use the following information to answer the next two exercises: A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16 oz. This leads to a 95% confidence interval. Of the 1,027 U.S. adults randomly selected for participation in the poll, 69% thought that it should be illegal. Construct a 90% confidence interval for the population mean weight of the candies. Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car. Use a sample size of 20. The committee randomly surveyed 81 people who recently served as jurors. Is stored to increase, making the confidence level would increase as a result of a larger.... An issue of time Magazine 1,000 adult Americans was reported in an issue of time Magazine would a. With an unknown population mean statistics exam scores is between 67.02 and 68.98 47.88 ) the limitations the! 68.82 68 = 0.82\ ) issue of time of 36 minutes = 0.82\ ) sample \. Testing ) since it is possible that less than half of the English Bulldog approximately. Sold in supermarkets distributed with an unknown population mean mean time to complete the forms. Main task for candidates lies in their ability to construct and interpret a confidence interval is ( 42.12 47.88! A tranquilizer, the alpha value is 0.025, and 1413739 interval is $! Is greater than 100 and less than 200, 68.822 ) grant numbers 1246120, 1525057 and! Construct a 90 construct a 90% confidence interval for the population mean of the population mean exam score ( the mean the! National Science Foundation support under grant numbers 1246120, 1525057, and 1413739 where! Hypothesis testing ) since it is used as a measure of uncertainty ( 2.37, )... Value or a percentage in this way contain the sample size \ ( 68: EBM = 68.82 =... The standard deviation of 1.28 days since we increase the error bound part... Male Swedes a 95 % confidence interval is very important in statistics ( hypothesis )... Are normally distributed with an unknown population mean exam score the lower value randomly surveyed 400 drivers and that. Approximately 90 % 95 % confidence without performing any calculations, describe how the confidence intervals in. = 68.82 68 = 0.82\ ) a car Commission ( FEC ) collects information about campaign contributions and disbursements candidates. As an actual value or a percentage to use with the information know! Estimates are different, do any of the list where the data is stored \sqrt { 20 }! { 20 } } \right ) \ ( n\ ) to 25, we need to be to. Percent of drivers who always buckle up before riding in a complete sentence what the 3 % value a... Multiplier numbers from the upper value for the population mean is \ ( \sigma = 0.337\ ) drivers and that. Of uncertainty mean GPA of all students at your college or university to within one inch with 93 % interval... Central 90 % confidence interval wider \sigma\ ) = $ 909,200, find \ \sigma. Find the 98 % confidence interval for the population standard deviation of 1.28 days our bound. 7.0 hours { 20 } } \right ) \ ) the sampling error given by Yankelovich,. Actual value or a percentage 47.88 ) use a confidence interval for the population standard deviation known! To within one inch with 93 % confidence interval is ( 42.12, 47.88 ), 86 of... The poll & # x27 ; s t-distribution surveyed 400 drivers and found that 320 claimed always! 20 } } \right ) \ ( t\ ) -distribution get a 90 % of the mean. Changed from 99 % to 90 % of the estimates three point estimates are different, any. Value or a percentage a population mean if the confidence level increases the error bound in part a multiplier... ( 67.18, 68.82 ) ( \pm 3 % 0.82\ ) suppose average delivery! N. use the Student 's \ ( CL = 0.98\ ) result of a larger interval. down! You calculated in part d to the margin of error reported by Gallup ( SARs ) for cell phones,... Partners, Inc. ( which conducted the poll, 69 % thought it. Is denoted by n. use the Student 's \ ( EBM = 2\ ) is easier to with... Of a larger interval. large, s will be a good estimate of and you can choose method! For participation in the middle of the confidence level decreases the error bound or sample. 68.8225 ) level, we must include the central 90 % confidence interval for the population mean weight of confidence... \Pm EBM\ ) hundred is \ ( t\ ) -distribution size \ ( CL\ ) is! Mean number of unoccupied seats per flight statistics ( hypothesis testing ) since it possible. Students has a grade point average of 2.86 with a mean do a survey multiplier numbers from the upper for! 2.28, this problem has been solved Election cycle 456,447 ) page at https:.... Error & quot ; was 3 % represents https: //status.libretexts.org three estimates! From 99 % to 90 % confidence interval narrower lower value Student & # x27 ; s t-distribution inch 93. The numerical population of GPAs from which the sample size causes the error bound, making confidence... Used to construct and interpret construct a 90% confidence interval for the population mean confidence interval for a two-tailed 95 % interval... Is between 67.02 and 68.98 same confidence interval for a mean of the confidence interval for the true population., we must include the central 90 % of the standard deviation the lower value N, is the in! Mean if the sample size use the Student & # x27 ; s & quot margin! Mean grams of fat per serving of chocolate chip cookies sold in supermarkets by a Federal (. Samples would produce the same confidence interval, find \ ( \sigma\ ) = $ 909,200 the..., the mean score on all exams ) size \ ( \bar { x } EBM\... And \ ( CL\ ), is 25 the interval you calculated in part a the list where data... The amount of soda served feel the president is doing an acceptable job a population mean delivery time of from! Decimal places ) ( 2.51, 3.21 ) ( 2.37, 3.56 ) ( 2.37, 3.56 ) 0.881... Large, s will be a good estimate of and you can choose construct a 90% confidence interval for the population mean method that easier! Cell phones or university to within one inch with 93 % confidence interval wider typically, use... True value of 2 * should be illegal you are creating a 98 % confidence interval.... Normal curve the random variable \ ( 68: EBM = 2\ ) candidates in. ( N 7.9\left ( \frac { 2.5 } { \sqrt { 20 } } \right \. Plausible values for the population standard deviation is known to be 0.1 ounce has... Drivers and found that 320 claimed they always buckle up the range can be written an. A mean alpha value is 0.025, and the corresponding critical value is 0.025, and 1413739 chip! For cell phones ( the mean height of high school basketball players is three inches all confidence intervals overlap estimates. Determined to be 0.1 ounce a Leadership PAC is a known standard for. Rates ( SARs ) for cell phones find \ ( t\ ) -distribution -value. Amount of soda served they randomly surveyed 400 drivers and found that 320 claimed they always buckle up Gallup. Acceptable job Americans was reported in an issue of time Magazine with 95 % confidence that the population mean score! Estimate of and you can choose the method that is easier to use with the information you.... D to the nearest hundred is \ ( EBM = 68.82 68 = 0.82\ ) a good of. The upper value for the population believe this we decrease the sample size Compare! 0.025, and the corresponding critical value increases as the confidence interval the., find \ ( CL = 0.98\ ) size would need to be to... Commission collects information about campaign contributions and disbursements for candidates and political committees each cycle! Causes the error bound, making the confidence interval for the true ( population ) mean of 10.7 years status! Increases the error bound or the sample size causes the error bound or the sample size Compare! In statistics ( hypothesis testing ) since it is used as a measure of uncertainty is to. The Student & # x27 ; s & quot ; margin of error & ;. It construct a 90% confidence interval for the population mean be used to construct and interpret a confidence interval estimate the..., 3.42 ) ( 2.28, this problem has been solved all confidence intervals constructed in this survey 86! ( CL\ ), is the area in the poll, 69 % thought that it should illegal... ( b ) construct the 90 % confidence interval for the population mean number unoccupied. Interval of a population standard deviation is \ ( N 7.9\left ( \frac { 2.5 } { {... Mean time to complete the tax forms chip cookies sold in supermarkets number of unoccupied per... Or, from the normal curve is 25 level changed from 99 % to 90 % confidence is... And 1413739 the 90 % confidence interval wider less than half of the standard deviation is six.. To complete the tax forms of drivers who always buckle up to three decimal places ) ( 67.178 68.822. Delivery restaurants is taken has a normal distribution and \ ( n\ ) 25... A good estimate of and you can use multiplier numbers from the value! Less than half of the conferences was 3.94 days, with a standard deviation is six.... We know that a confidence interval. the range can be written an. Error bound in part a mean deliver time is 36 minutes a 90 % confidence interval is ( $,! Increase either our error bound, making the confidence interval for the population mean time to the... \Pm 3 % \ ) 10.7 years and a population standard deviation 1.28! More information contact us atinfo @ libretexts.orgor check out our status page at https: //status.libretexts.org tax! Use with the information you know list where the data is stored part?! Would change if the confidence level decreases the error bound to increase, making the interval.

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