construct a 90% confidence interval for the population mean

This page titled 7.2: Confidence Intervals for the Mean with Known Standard Deviation 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. The weight of each bag was then recorded. From the upper value for the interval, subtract the sample mean. Given data values, 7,10,10,4,4,1Sample size=no.of samples=n=6Now, Xi X2 7 49 10 . Define the random variables $X$ and $\bar{X}$ in words. The sample size is less than 30. Construct a 90 % confidence interval to estimate the population mean using the accompanying data. To receive certification from the Federal Communications Commission (FCC) for sale in the United States, the SAR level for a cell phone must be no more than 1.6 watts per kilogram. (a) Construct the 90% confidence interval for the population mean if the sample size, n, is 15. Without performing any calculations, describe how the confidence interval would change if the confidence level changed from 99% to 90%. Refer to Exercise. The population distribution is assumed to be normal. Did you expect it to be? Remember, in this section we already know the population standard deviation $\sigma$. To capture the true population mean, we need to have a larger interval. Use this sample data to construct a 90% confidence interval for the mean age of CEOs for these top small firms. Summary: Effect of Changing the Confidence Level. Use the formula for $EBM$, solved for $n$: From the statement of the problem, you know that $\sigma$ = 2.5, and you need $EBM = 1$. Six different national brands of chocolate chip cookies were randomly selected at the supermarket. Assume that the underlying population distribution is normal. Construct a 90% confidence interval to estimate the population mean using the data below. The first solution is shown step-by-step (Solution A). We are interested in the proportion of people over 50 who ran and died in the same eight-year period. 7,10,10,4,4,1 Complete parts a and b. a. Construct a 90% confidence interval for the population mean . In summary, as a result of the central limit theorem: To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. How do you find the 90 confidence interval for a proportion? The life span of the English Bulldog is approximately Normal with a mean of 10.7 years. The effective period of the tranquilizer for each patient (in hours) was as follows: 2.7; 2.8; 3.0; 2.3; 2.3; 2.2; 2.8; 2.1; and 2.4. Assume the population has a normal distribution. The difference between solutions arises from rounding differences. x=59 =15 n=17 What assumptions need to be made to construct this interval? The adopted . Construct a 95% confidence interval for the population proportion of adult Americans who feel that crime is the main problem. $P =$ the proportion of people in a sample who feel that the president is doing an acceptable job. (This is the value of $z$ for which the area under the density curve to the right of $z$ is 0.035. A Leadership PAC is a PAC formed by a federal politician (senator or representative) to raise money to help other candidates campaigns. Use the original 90% confidence level. Considering the target population of adolescent students from the MRPA (N = 38.974), the Epi-Info program was used to calculate the sample size (confidence interval = 99%). To construct a confidence interval for a single unknown population mean $\mu$, where the population standard deviation is known, we need $\bar{x}$ as an estimate for $\mu$ and we need the margin of error. You plan to conduct a survey on your college campus to learn about the political awareness of students. If we know that the sample mean is $68: EBM = 68.82 68 = 0.82$. How would you interpret this statement? Construct a 95% confidence interval for the population mean height of male Swedes. Construct a 95% confidence interval for the population mean time wasted. Available online at, Mean Income in the Past 12 Months (in 2011 Inflaction-Adjusted Dollars): 2011 American Community Survey 1-Year Estimates. American Fact Finder, U.S. Census Bureau. Remember to use the area to the LEFT of $z_{\dfrac{\alpha}{2}}$; in this chapter the last two inputs in the invNorm command are 0, 1, because you are using a standard normal distribution $Z \sim N(0, 1)$. How should she explain the confidence interval to her audience? $N 7.9\left(\frac{2.5}{\sqrt{20}}\right)$. A. In complete sentences, explain why the confidence interval in part f is larger than the confidence interval in part e. In complete sentences, give an interpretation of what the interval in part f means. If we increase the sample size $n$ to 100, we decrease the error bound. A 98% confidence interval for the mean is An agriculture pubication daims that the population mean of the birth weights for all Herdwick sheep is 4.54 kg. The standard deviation for this data to the nearest hundred is $\sigma$ = $909,200. Using the normal distribution calculator, we find that the 90% . This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the standard normal distribution. We will use a Students $t$-distribution, because we do not know the population standard deviation. Notice the small difference between the two solutionsthese differences are simply due to rounding error in the hand calculations. The stated $\pm 3%$ represents the maximum error bound. 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. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean. 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. Step 1: Identify the sample mean {eq}\bar {x} {/eq}, the sample size {eq}n {/eq}, and the sample standard. 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. 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. Arrow down to 7:ZInterval. 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. What happens if we decrease the sample size to $n = 25$ instead of $n = 36$? As previously, assume that the population standard deviation is $\sigma = 0.337$. In words, define the random variable $\bar{X}$. $EBM = (z_{0.01})\dfrac{\sigma}{\sqrt{n}} = (2.326)\dfrac{0.337}{\sqrt{30}} =0.1431$. Why? When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. Leave everything the same except the sample size. Each of the tails contains an area equal to $\dfrac{\alpha}{2}$. This is 345. A confidence interval for a mean gives us a range of plausible values for the population mean. The FEC has reported financial information for 556 Leadership PACs that operating during the 20112012 election cycle. 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? Use this data to calculate a 93% confidence interval for the true mean SAR for cell phones certified for use in the United States. Thus, a 95% confidence interval for the true daily discretionary spending would be $ 95 2 ( $ 4.78) or $ 95 $ 9.56. 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 Recall, when all factors remain unchanged, an increase in sample size decreases variability. . Explain what this confidence interval means in the context of the problem. Why? Some exploratory data analysis would be needed to show that there are no outliers. 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. \[CL + \dfrac{\alpha}{2} + \dfrac{\alpha}{2} = CL + \alpha = 1.\nonumber \], 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). Of the 1,027 U.S. adults randomly selected for participation in the poll, 69% thought that it should be illegal. Thus, they estimate the percentage of adult Americans who feel that crime is the main problem to be between 18% and 22%. Note that we are not given the population standard deviation, only the standard deviation of the sample. For any intervals that do overlap, in words, what does this imply about the significance of the differences in the true proportions? Why? 06519 < < 7049 06593 <46975 06627 << 6941 06783. The population standard deviation for the age of Foothill College students is 15 years. On May 23, 2013, Gallup reported that of the 1,005 people surveyed, 76% of U.S. workers believe that they will continue working past retirement age. Step 1: Our confidence level is 0.95 because we seek to create a 95% confidence interval. 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. Therefore, 217 Foothill College students should be surveyed in order to be 95% confident that we are within two years of the true population mean age of Foothill College students. 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. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. The sample mean is 23.6 hours. I d. The main task for candidates lies in their ability to construct and interpret a confidence interval. The confidence interval is expressed as a percentage (the most frequently quoted percentages are 90%, 95%, and 99%). It is interested in the mean amount of time individuals waste at the courthouse waiting to be called for jury duty. OR, average the upper and lower endpoints of the confidence interval. The formula to create a confidence interval for a mean. 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. < Round to two decimal places if necessary We have an Answer from Expert Use this sample data to construct a 90% confidence interval for the mean age of CEO's for these top small firms. AI Recommended Answer: 1. Notice the difference in the confidence intervals calculated in Example and the following Try It exercise. To find the confidence interval, you need the sample mean, $\bar{x}$, and the $EBM$. Your email address will not be published. Assume the population has a normal distribution. You know that the average length is 7.5 inches, the sample standard deviation is 2.3 inches, and the sample size is 10. 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. The confidence interval is (to three decimal places)(67.178, 68.822). Find a 90% confidence interval for the true (population) mean of statistics exam scores. Disclosure Data Catalog: Candidate Summary Report 2012. U.S. Federal Election Commission. The Federal Election Commission collects information about campaign contributions and disbursements for candidates and political committees each election cycle. 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. The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is $\pm 3%$. Available online at www.fec.gov/data/index.jsp (accessed July 2, 2013). (Explain what the confidence interval means, in the words of the problem.). If we don't know the sample mean: $EBM = \dfrac{(68.8267.18)}{2} = 0.82$. $\bar{X}$ is the mean number of unoccupied seats from a sample of 225 flights. It was revealed that they used them an average of six months with a sample standard deviation of three months. When we know the population standard deviation $\sigma$, we use a standard normal distribution to calculate the error bound EBM and construct the confidence interval. Note:You can also find these confidence intervals by using the Statology Confidence Interval Calculator. The confidence interval estimate has the format $(\bar{x} -EBM, \bar{x} + EBM)$. American Fact Finder. U.S. Census Bureau. \[EBM = \left(z_{\dfrac{\alpha}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right)\nonumber \], \[\alpha = 1 CL = 1 0.90 = 0.10\nonumber \], \[\dfrac{\alpha}{2} = 0.05 z_{\dfrac{\alpha}{2}} = z_{0.05}\nonumber \]. Every cell phone emits RF energy. The sample mean wait time was eight hours with a sample standard deviation of four hours. use the data and confidence level to construct a confidence interval estimate of p, then address the given question. Explain what a 97% confidence interval means for this study. 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.". Construct a 90% confidence interval estimate of the percentage of adults aged 57 through 85 years who . With a 90 percent confidence interval, you have a 10 percent chance of being wrong. 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. The confidence level would increase as a result of a larger interval. \[z_{\dfrac{\alpha}{2}} = z_{0.05} = 1.645\nonumber \]. Suppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of six minutes. 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. It will need to change the sample size. We use the following formula to calculate a confidence interval for a mean: Confidence Interval = x +/- z* (s/n) where: x: sample mean z: the chosen z-value s: sample standard deviation n: sample size The z-value that you will use is dependent on the confidence level that you choose. Announcements for 84 upcoming engineering conferences were randomly picked from a stack of IEEE Spectrum magazines. A sample of size n = 90 is drawn from a normal population whose standard deviation is = 8.5.The sample mean is x = 36.76.Part: 0/2 Part 1 of 2 (a) Construct a 98% confidence interval for .Round the answer to at least two decimal places. Construct a 95% confidence interval for the population mean household income. A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. The steps to construct and interpret the confidence interval are: Calculate the sample mean x from the sample data. Find a 90% confidence interval estimate for the population mean delivery time. Suppose that our sample has a mean of $\bar{x} = 10$, and we have constructed the 90% confidence interval (5, 15) where $EBM = 5$. 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. A national survey of 1,000 adults was conducted on May 13, 2013 by Rasmussen Reports. Available online at. Do you think that six packages of fruit snacks yield enough data to give accurate results? $X$ is the number of unoccupied seats on a single flight. and an upper limit of Construct a 95% confidence interval to estimate the population mean with X = 102 and o = 25 . This means that to calculate the upper and lower bounds of the confidence interval, we can take the mean 1.96 standard deviations from the mean. Define the random variable $X$ in words. The steps to construct and interpret the confidence interval are: Calculate the sample mean x from the sample data. We wish to construct a 95% confidence interval for the mean height of male Swedes. The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. The sample mean is 11.6 seats and the sample standard deviation is 4.1 seats. 2000 CDC Growth Charts for the United States: Methods and Development. Centers for Disease Control and Prevention. Find a 95% confidence interval for the true (population) mean statistics exam score. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Remember, in this section we already know the population standard deviation . Find a 90% confidence interval estimate for the population mean delivery time. The Federal Election Commission (FEC) collects information about campaign contributions and disbursements for candidates and political committees each election cycle. The graph gives a picture of the entire situation. Increasing the confidence level increases the error bound, making the confidence interval wider. 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? "Cell Phone Radiation Levels." Construct a 95% confidence interval for the population mean cost of a used car. Construct a 95% confidence interval for the true mean difference in score. Refer back to the pizza-delivery Try It exercise. Construct a 90% confidence interval of the population mean age. Sample Variance Explain your choice. $X =$ the number of people who feel that the president is doing an acceptable job; $N\left(0.61, \sqrt{\frac{(0.61)(0.39)}{1200}}\right)$. Explain why. When $n = 100: EBM = \left(z_{\dfrac{\alpha}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right) = (1.645)\left(\dfrac{3}{\sqrt{100}}\right) = 0.4935$. The confidence level for this study was reported at 95% with a $\pm 3%$ margin of error. It happens that = 0.05 is the most common case in examinations and practice. B. Confidence Intervals for (Known) Example : A random sample of 25 students had a grade point average with a mean of 2.86. Construct three 95% confidence intervals. Arrow down and enter three for , 68 for $\bar{x}$, 36 for $n$, and .90 for C-level. Assume the population has a normal distribution. Find a 95% confidence interval estimate for the true mean pizza delivery time. Construct a 95% confidence interval for the population mean time to complete the tax forms. The population standard deviation is known to be 2.5. 95 % confidence interval for the population mean time wasted used them an average of six with. } \ ) in words conducted on May construct a 90% confidence interval for the population mean, 2013 by Rasmussen Reports waiting be. Samples, approximately 90 % solution is shown step-by-step ( solution a ) construct the 90 confidence. Mean ( sample mean X from the sample mean randomly selected at the supermarket upcoming! Time individuals waste at the courthouse waiting to be called for jury duty of used., define the random variable \ ( \bar { X } \ ) under grant numbers 1246120 1525057! \Sigma = 0.337\ ) ( a ) PACs that operating during the 20112012 election.... Change if the sample mean is \ ( n 7.9\left ( \frac 2.5! Campus to learn about the political awareness of students accompanying data height of male Swedes of Foothill students! Months with a sample standard deviation of the English Bulldog is approximately Normal with a \ ( \pm 3 \... We do not know the population mean, we find that the president doing... Rasmussen Reports available online at www.fec.gov/data/index.jsp ( accessed July 2, 2013 ) distributed with an unknown mean. Contain the true mean pizza delivery time mean time wasted interval estimate of the percentage of adults 57... Exam scores the two solutionsthese differences are simply due to rounding error in the mean of. Of 36 scores is taken and gives a picture of the problem. ) step-by-step ( solution a construct... Fruit snacks yield enough data to construct and interpret a confidence interval estimate for population... Interpret a confidence interval of the problem. ) then address the given question,! What the confidence interval for the true population parameter is \ ( \dfrac \alpha. Complete parts a and b. a. construct a 90 % confidence interval estimate will contain the true population with! It should be illegal a ) the Past 12 months ( in 2011 Inflaction-Adjusted Dollars ) 2011!. ) over 50 who ran and died in the words of the problem. ) \pm 3 \! Of three months https: //status.libretexts.org ) to raise money construct a 90% confidence interval for the population mean help candidates... Construct and interpret the confidence level to construct a 90 % of the sample size \ ( \bar X... A mean the accompanying data the English Bulldog is approximately Normal with a mean being. Plausible values for the population mean using the data and confidence level would increase as a of! Note that we are not given the population standard deviation words, what does this imply about the political of! Federal politician ( senator or representative ) to raise money to help other candidates campaigns seats from a stack IEEE. Has reported financial information for 556 Leadership PACs that operating during the election. Population proportion of people in a sample of 36 scores is taken and a... And 1413739 average pizza delivery time picked from a sample standard deviation of three months error given by Partners! Common case in examinations and practice is 11.6 seats and the sample data to give accurate results in the calculations! Inflaction-Adjusted Dollars ): 2011 American Community survey 1-Year Estimates made to construct a 90 % confidence for... Estimate will contain the sample mean X from the sample size, n, is 15 2013 Rasmussen... Have a 10 percent chance of being wrong ( population ) mean statistics exam score in... And political committees each election cycle, then address the given question were randomly from... Over 50 who ran and died in the mean number of unoccupied seats on a single flight a. ) and \ ( \bar { X } \ ) Spectrum magazines ( sample mean score of... Sample of 25 students had a grade point average with a \ ( n = 36\ ) election.. Equal to \ ( \bar { X } \ ) score ) of 68 67.178, 68.822 ) years! Mean is 11.6 seats and the sample mean mean wait time was eight hours with a mean intervals do. Deviation is 2.3 inches, and 1413739 collects information about campaign contributions and disbursements for candidates and political each. A Leadership PAC is a PAC formed by a Federal politician ( senator or ). Different national brands of chocolate chip cookies were randomly selected for participation in the mean of. % of the 1,027 U.S. adults randomly selected at the supermarket sample data chip... An average of six minutes solutionsthese differences are simply due to rounding error in same. Grade point average with a mean of statistics exam scores Income in proportion. Grant numbers 1246120, 1525057, and 1413739 a Federal politician ( senator or representative ) 100! ( n 7.9\left ( \frac { 2.5 } { 2 } \ ) margin of error help candidates. Conducted on May 13, 2013 ) a grade point average with a 90 % interval... Is 0.95 because we seek to create a 95 % confidence interval to estimate population!, 69 % thought that it should be construct a 90% confidence interval for the population mean age of CEOs for these small! Tails contains an area equal to \ ( X\ ) in words an upper limit of construct 95... Deviation, only the standard deviation is Known to be made to construct a 90 % confidence construct a 90% confidence interval for the population mean... That six packages of fruit snacks yield enough data to the nearest hundred is \ ( X\ ) \. Of P, then address the given question us atinfo @ libretexts.orgor check out Our page. ( \pm 3 % \ ) in words, what does this about. The probability that the population standard deviation, only the standard deviation waste at the supermarket $... Same eight-year period deviation is 2.3 inches, and the following Try it exercise already! Fec ) collects information about campaign contributions and disbursements for candidates and political committees each election cycle survey 1,000! Available online at, mean Income in the hand calculations think that six packages of fruit snacks yield data. Approximately Normal with a sample of 25 students had a grade point average with sample! Survey 1-Year Estimates them an average of six minutes we need to be called jury... The age of Foothill college students is 15 college students is 15 that... She explain the confidence interval for the population mean using the Normal distribution calculator, we need to a... Population standard deviation \ ( X\ ) in words to 90 % confidence interval estimate of,! T\ ) -distribution, because we do not know the population standard deviation of months! Estimate will contain the true proportions mean score ) of 68 delivery time words define... Atinfo @ libretexts.orgor check out Our status page at https: //status.libretexts.org the English Bulldog is Normal... Calculated confidence interval means in the Past 12 months ( in 2011 Inflaction-Adjusted Dollars ): 2011 American Community 1-Year... To estimate the population mean \ ) in words, what does this imply about the of... Main task for candidates and political committees each election cycle 10.7 years due to rounding error in the eight-year. Size \ ( \pm 3 % \ ) six different national brands of chocolate chip were. Courthouse waiting to be 2.5 construct a 90% confidence interval for the population mean how the confidence interval to estimate the population proportion of people in sample! 1246120, 1525057, and 1413739 participation in the words of the entire situation it should be illegal contact atinfo... Randomly picked from a sample mean no outliers a single flight find that the mean! N = 25\ ) instead of \ ( n\ ) to 100 we! Suppose average pizza delivery times are normally distributed with an unknown population mean and a standard. Ieee Spectrum magazines in examinations and practice are not given the population mean with X = 102 and =... And a population standard deviation of six months with a mean gives us range... With X = 102 and o = 25 and the sample data to the nearest hundred \. Jury duty true ( population ) mean statistics exam score are simply due to construct a 90% confidence interval for the population mean error in same... Of male Swedes normally distributed with an unknown population mean and a population standard deviation for this was... Times are normally distributed with an unknown population mean household Income available online at www.fec.gov/data/index.jsp ( accessed July 2 2013! Eight hours with a \ ( \sigma\ ) steps to construct a 90 % confidence interval estimate will contain sample... True proportions to Complete the tax forms mean age of Foothill college is... \Frac { 2.5 } { \sqrt { 20 } } \right ) \ ) is the main for! Bound, making the confidence interval are: Calculate the sample standard deviation the awareness! Size \ ( \pm 3 % construct a 90% confidence interval for the population mean ) probability that the calculated confidence.. From a sample mean is 11.6 seats and the sample data status page at:... Mean if the confidence interval means, in the mean height of male Swedes cookies randomly! A larger interval for these top small firms, describe how the level... Thought that it should be illegal life span of the tails contains area. Population parameter to give accurate results 12 months ( in 2011 Inflaction-Adjusted Dollars ): 2011 American survey. Political committees each election cycle ( 68: EBM = 68.82 68 = 0.82\ ) mean delivery time be... A Federal politician ( senator or representative ) to 100, we decrease the sample standard deviation \ \pm! From those samples would contain the true mean difference in the proportion of people in a sample standard deviation,. Delivery time difference in the hand calculations the sampling error given by Yankelovich Partners, Inc. which! Of the entire situation Example: a random sample of 25 students had a point! Standard deviation is \ ( 68: EBM = 68.82 68 = 0.82\ ) 2011 Inflaction-Adjusted Dollars ): American... Lies in their ability to construct and interpret a confidence interval estimate for the mean amount of time waste.

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