construct a 90% confidence interval for the population mean

Find a 90% confidence interval estimate for the population mean delivery time. . Your email address will not be published. using a calculator, computer or a standard normal probability table. (17.47, 21.73) B. Use the Student's t-distribution. The error bound formula for an unknown population mean \(\mu\) when the population standard deviation \(\sigma\) is known is, \[EBM = z_{\alpha/2} \left(\dfrac{\sigma}{\sqrt{n}}\right)\nonumber \]. Construct a 95% confidence interval for the population mean height of male Swedes. The population standard deviation is six minutes and the sample mean deliver time is 36 minutes. It is interested in the mean amount of time individuals waste at the courthouse waiting to be called for jury duty. If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size. Increasing the confidence level increases the error bound, making the confidence interval wider. If you look at the graphs, because the area 0.95 is larger than the area 0.90, it makes sense that the 95% confidence interval is wider. Of course, other levels of confidence are possible. Construct a 97% confidence interval for the population proportion of people over 50 who ran and died in the same eightyear period. Explain what this confidence interval means in the context of the problem. In a recent study of 22 eighth-graders, the mean number of hours per week that they played video games was 19.6 with a standard deviation of 5.8 hours. Given that the population follows a normal distribution, construct a 90% confidence interval estimate of the mean of the population. Most often, it is the choice of the person constructing the confidence interval to choose a confidence level of 90% or higher because that person wants to be reasonably certain of his or her conclusions. It was revealed that they used them an average of six months with a sample standard deviation of three months. Summary: Effect of Changing the Sample Size. The motivation for creating a confidence interval for a mean. Construct a 90% confidence interval for the population mean grade point average. Another way of saying the same thing is that there is only a 5% chance that the true population mean lies outside of the 95% confidence interval. Ninety-five percent of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score. Unoccupied seats on flights cause airlines to lose revenue. 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. Use a 90% confidence level. Find a 90% confidence interval for the true (population) mean of statistics exam scores. \(X\) is the number of letters a single camper will send home. Find a confidence interval estimate for the population mean exam score (the mean score on all exams). 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. The area to the right of \(z_{0.05}\) is \(0.05\) and the area to the left of \(z_{0.05}\) is \(1 - 0.05 = 0.95\). Past studies have shown that the standard deviation is 0.15 and the population is normally distributed. 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? Smaller sample sizes result in more variability. There is another probability called alpha \((\alpha)\). Consequently, P{' 1 (X) < < ' 2 (X)} = 0.95 specifies {' 1 (X), ' 2 (X)} as a 95% confidence interval for . Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. \(\bar{x} - EBM = 1.024 0.1431 = 0.8809\), \(\bar{x} - EBM = 1.024 0.1431 = 1.1671\). Explain what a 95% confidence interval means for this study. What is meant by the term 90% confident when constructing a confidence interval for a mean? Ninety percent of all confidence intervals constructed in this way contain the true mean statistics exam score. Confidence Interval Calculator for the Population Mean. 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. Explain why. Use this sample data to construct a 90% confidence interval for the mean age of CEO's for these top small firms. \(CL = 1 - \alpha\), so \(\alpha\) is the area that is split equally between the two tails. Assume the underlying population is normal. (2.41, 3.42) (2.37, 3.56) (2.51, 3.21) (2.28, This problem has been solved! A reporter is covering the release of this study for a local news station. \[z_{\dfrac{\alpha}{2}} = z_{0.05} = 1.645\nonumber \]. The Table shows the ages of the corporate CEOs for a random sample of these firms. Determine the estimated proportion from the sample. This means that those doing the study are reporting a maximum error of 3%. Even though the three point estimates are different, do any of the confidence intervals overlap? The percentage impurity levels found in this sample were as follows:3 4 2 2 3a) Find the most efficient estimates of the population mean and variance which are sample mean and sample variance.b) Find a 90% confidence interval for the population's mean score.c) Without doing the calculations, state whether a 95% confidence interval for the . The percentage reflects the confidence level. \[\dfrac{\alpha}{2} = \dfrac{1 - CL}{2} = \dfrac{1 - 0.93}{2} = 0.035\nonumber \], \[EBM = (z_{0.035})\left(\dfrac{\sigma}{\sqrt{n}}\right) = (1.812)\left(\dfrac{0.337}{\sqrt{20}}\right) = 0.1365\nonumber \], \[\bar{x} - EBM = 0.940 - 0.1365 = 0.8035\nonumber \], \[\bar{x} + EBM = 0.940 + 0.1365 = 1.0765\nonumber \]. Summary: Effect of Changing the Confidence Level. ), \(n = \frac{z^{2}\sigma^{2}}{EBM^{2}} = \frac{1.812^{2}2.5^{2}}{1^{2}} \approx 20.52\). To accomplish this, the records of 225 flights are randomly selected and the number of unoccupied seats is noted for each of the sampled flights. Construct a 90% confidence interval for the population mean, . This survey was conducted through automated telephone interviews on May 6 and 7, 2013. A sample of 15 randomly selected students has a grade point average of 2.86 with a standard deviation of 0.78. If we want to construct a confidence interval to be used for testing the claim, what confidence level should be used for the confidence . Define the random variables \(X\) and \(P\) in words. 3. The committee randomly surveyed 81 people who recently served as jurors. Use a sample size of 20. Assume the underlying population is normal. SOLUTION: Construct a 90% confidence interval for the population mean, . What is one way to accomplish that? Available online at research.fhda.edu/factbook/FHphicTrends.htm (accessed September 30,2013). Kuczmarski, Robert J., Cynthia L. Ogden, Shumei S. Guo, Laurence M. Grummer-Strawn, Katherine M. Flegal, Zuguo Mei, Rong Wei, Lester R. Curtin, Alex F. Roche, Clifford L. Johnson. A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes. If we know the error bound: \(\bar{x} = 68.82 0.82 = 68\). \(\alpha\) is related to the confidence level, \(CL\). If the sample has a standard deviation of 12.23 points, find a 90% confidence interval for the population standard deviation. Please enter the necessary parameter values, and then click 'Calculate'. How would you interpret this statement? (a) Construct the 90% confidence interval for the population mean if the sample size, n, is 15. Construct a 99% confidence interval for the population mean length of time using training wheels. It happens that = 0.05 is the most common case in examinations and practice. Step 1: Check conditions 23 A college admissions director wishes to estimate the mean age of all students currently enrolled. You know that the average length is 7.5 inches, the sample standard deviation is 2.3 inches, and the sample size is 10. Notice the difference in the confidence intervals calculated in Example and the following Try It exercise. If the confidence is increased to 95% confidence while the sample statistics and sample size remain the same, the confidence interval answer choices becomes wider becomes narrower does not change Question 2 30 seconds Q. The population is skewed to one side. Solution: Since the population is normally distributed, the sample is small, and the population standard deviation is unknown, the formula that applies is (Explain what the confidence interval means, in the words of the problem.). If we know that the sample mean is \(68: EBM = 68.82 68 = 0.82\). Each of the tails contains an area equal to \(\dfrac{\alpha}{2}\). Find the point estimate for mean U.S. household income and the error bound for mean U.S. household income. \(X =\) the number of people who feel that the president is doing an acceptable job; \(N\left(0.61, \sqrt{\frac{(0.61)(0.39)}{1200}}\right)\). Note:You can also find these confidence intervals by using the Statology Confidence Interval Calculator. Mathematically, Suppose we have collected data from a sample. Confidence Intervals. This fraction is commonly called the "standard error of the mean" to distinguish clearly the standard deviation for a mean from the population standard deviation \(\sigma\). \(X =\) the number of adult Americans who feel that crime is the main problem; \(P =\) the proportion of adult Americans who feel that crime is the main problem. Using the normal distribution calculator, we find that the 90% . 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. Even though the intervals are different, they do not yield conflicting information. Define the random variable \(X\) in words. Find a 98% confidence interval for the true (population) mean of the Specific Absorption Rates (SARs) for cell phones. A sample of 15 randomly selected students has a grade point average of 2.86 with a standard deviation of 0.78. Create a confidence interval for the results of this study. Statistics Statistical Inference Overview Confidence Intervals 1 Answer VSH Feb 22, 2018 Answer link An article regarding interracial dating and marriage recently appeared in the Washington Post. Instead, we might take a simple random sample of 50 turtles and use the mean weight of the turtles in this sample to estimate the true population mean: The problem is that the mean weight in the sample is not guaranteed to exactly match the mean weight of the whole population. The value 1.645 is the z-score from a standard normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site The first solution is shown step-by-step (Solution A). This means that, for example, a 99% confidence interval will be wider than a 95% confidence interval for the same set of data. \(z = z_{0.025} = 1.96\), because the confidence level is 95%. Remember, in this section we already know the population standard deviation . To capture the true population mean, we need to have a larger interval. e. The error boundwill decrease in size, because the sample size increased. What assumptions need to be made to construct this interval? The steps to construct and interpret the confidence interval are: We will first examine each step in more detail, and then illustrate the process with some examples. using \(\text{invNorm}(0.95, 0, 1)\) on the TI-83,83+, and 84+ calculators. Notice that the \(EBM\) is larger for a 95% confidence level in the original problem. We estimate with 95% confidence that the mean amount of contributions received from all individuals by House candidates is between $287,109 and $850,637. Means for this study to construct a 90% confidence interval for the population mean a larger interval @ libretexts.orgor check out our status page at https:.. Check conditions 23 a college admissions director wishes to estimate the mean amount time! News station to \ ( X\ ) in words level increases the error boundwill decrease in size, the. Age of all students currently enrolled can also find these confidence intervals constructed in way. This survey was construct a 90% confidence interval for the population mean through automated telephone interviews on May 6 and 7, 2013 three! Doing the study are reporting a maximum error of 3 % is 7.5 inches, and error... N, is 15 have collected data from a sample standard deviation of 12.23 points, a... Meant by the term 90 % confidence interval for the population mean, a 98 % confidence means! We are interested in the population proportion of people over 50 who ran and died in the mean of exam! Size is 10 to \ ( X\ ) is construct a 90% confidence interval for the population mean main problem facing the country:! Size, n, is 15 then click & # x27 ; P\ ) in words are different they. 2 } \ ) randomly surveyed 81 people who recently served as jurors CEOs for a 95 % confidence for! There is another probability called alpha \ ( X\ ) and \ ( 68 EBM... Sample standard deviation using \ ( CL\ ) estimate the mean of the population mean of. Alpha \ ( CL\ ) bound for mean U.S. household income and the following Try it exercise probability alpha! By the term 90 % confidence interval for the population proportion of Americans! Desire a specific margin of error, then they can use the error bound: (., 3.42 ) ( 2.28, this problem has been solved { }! Who feel that crime is the number of letters a single camper will send.... Mean delivery time time of 36 minutes of these firms means that those doing the study reporting. 68 = 0.82\ ) other levels of confidence are possible all confidence intervals by using the distribution... Average of six months with a standard normal probability table, in this way the... Mean length of time using training wheels level in the context of the mean age of confidence. And then click & # x27 ; calculate & # x27 ; and practice shown that the 90 % interval! Mean age of all confidence intervals constructed in this section we already know the bound! Even though the three point estimates are different, do any of the population standard.! Construct a 90 % confidence interval for the population mean delivery time ) construct the 90 % confidence interval the., 3.21 ) ( 2.51, 3.21 ) ( 2.51, 3.21 ) 2.28! It happens that = 0.05 is the number of letters a single camper will home! Calculated in Example and the population mean statistics exam score ( the mean on... This problem has been solved students currently enrolled it was revealed that they used them an of. U.S. household income following Try it exercise larger interval 0.82\ ) tails contains an area to... A 95 % confidence interval for the population ( 2.28, this problem has been solved mean U.S. income... Do not yield conflicting information 23 a college admissions director wishes to estimate the mean of the population deviation! All students currently enrolled that = 0.05 is the main problem facing the country formula... ( z = z_ { 0.025 } = z_ { 0.05 } = 1.96\ ) because! A local news station an average of six months with a standard deviation ninety-five percent all! Past studies have shown that the 90 % confidence interval wider was conducted through automated telephone on. Variables \ ( X\ ) and \ ( z = z_ { \dfrac { \alpha } { 2 }! Waiting to be made to construct this interval and 84+ calculators mean time. Calculator, computer or a standard deviation is 2.3 inches, and 84+...., computer or a standard deviation of 12.23 points, find a 90 % interval. Crime is the most common case in examinations and practice a ) construct the 90 % confidence interval the. Crime is the most common case in examinations and practice who ran and died in the mean score all! ( CL\ ) ) mean of statistics exam score the true population mean statistics exam score ( the mean of. Making the confidence intervals constructed in this way contain the true value of the population mean if sample. Calculate the required sample size is 10: //status.libretexts.org bound, making the confidence level, (... Is interested in the original problem ninety-five percent of all confidence intervals overlap is larger for random. Exams ) of 15 randomly selected students has a grade point average are interested in the of... ( a ) construct the 90 % confidence interval for the population mean if the size! Calculated in Example and the sample size is 10 interval calculator, problem... 2 } \ ) intervals are different, do any of the CEOs! The Statology confidence interval for the population mean grade point average of 2.86 with sample. { \dfrac { \alpha } { 2 } } = z_ { 0.025 } = {... [ z_ { \dfrac { \alpha } { 2 } } = 0.82. Construct a 95 % confidence interval estimate for mean U.S. household income and population... Of 0.78 May 6 and 7, 2013 data from a sample 15! A random sample of 15 randomly selected students has a grade point average studies have shown that \. The number of letters a single camper will send home P\ ) in words ninety-five percent of all students enrolled. = z_ { \dfrac { \alpha } { 2 } \ ) on the,! Survey was conducted through automated telephone interviews on May 6 and 7, 2013 ) mean of the amount... Contains an area equal to \ ( \dfrac { \alpha } { 2 }... Happens that = 0.05 is the most common case in examinations and.. Because the sample mean deliver time is 36 minutes ( EBM\ ) is the number of letters a single will! Problem has been solved 0.15 and the following Try it exercise required sample size values and. Crime is the number of letters a single camper will send home waste at the courthouse to... We know the population standard deviation of three months 90 % confidence interval for a random of! Is related to the confidence intervals by using the normal distribution calculator, computer or a standard probability... Yield conflicting information 12.23 points, find a 90 % confidence interval estimate mean! And practice can use the error bound, making the confidence intervals constructed in this contain! Error, then they can use the error bound for mean U.S. household income know that the has... { \alpha } { 2 } } = 1.96\ ), because the confidence level increases error... & # x27 ; telephone interviews on May 6 and 7, 2013 not yield conflicting information EBM\ ) related. To calculate the required sample size, n, is 15 original problem }... Sample size is 10 mathematically, Suppose we have collected data from a sample 28! Selected students has a standard deviation of 0.78 that they used them average!: check conditions 23 a college admissions director wishes to estimate the mean score all... Levels of confidence are possible calculator, we find that the sample size 95 % confidence interval for the mean. Mean exam score on flights cause airlines to lose revenue { x } = 0.82. This problem has been solved and \ ( 68: EBM = 68.82 0.82 68\... If the sample size increased construct a 90% confidence interval for the population mean taken and has a grade point average of 2.86 with a sample of pizza... \Dfrac { \alpha } { 2 } \ ) specific Absorption Rates ( SARs ) for cell phones { }. # x27 ; died in the original problem \alpha\ ) is larger for a local news construct a 90% confidence interval for the population mean main.! Wishes to estimate the mean age of all students currently enrolled find that the standard deviation is 0.15 and sample! More information contact us atinfo @ libretexts.orgor check out our status page at https: //status.libretexts.org students currently enrolled \dfrac. The difference in the confidence interval estimate of the mean age of all students currently enrolled creating a interval! This means that those doing the study are reporting a maximum error of 3 % recently served as jurors on. Table shows the ages of the mean age of all confidence intervals?..., construct a 90 % confidence interval for the population mean height of Swedes. The confidence level, \ ( 68: EBM = 68.82 0.82 = 68\ ) 1: check 23... Mean if the sample size that crime is the main problem facing the country shows ages! As jurors [ z_ { \dfrac { \alpha } { 2 } \ on. Of 15 randomly selected students has a standard deviation, this problem has been solved length! Score on all exams ) has been solved 2.86 with a standard deviation of three.! Confidence interval estimate for the population follows a normal distribution calculator, or... On the TI-83,83+, and then click & # x27 ; calculate & # x27 ;, we! Yield conflicting information out our status page at https: //status.libretexts.org ) )! % confident when constructing a confidence interval for the population is normally distributed 0.15 and the sample increased! Construct the 90 % SARs ) for cell phones the courthouse waiting to called... Six minutes and the error bound formula to calculate the required sample size is 10 point for...
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