construct a 90% confidence interval for the population mean

The weight of each bag was then recorded. For a two-tailed 95% confidence interval, the alpha value is 0.025, and the corresponding critical value is 1.96. 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. The sample size is less than 30. The random sample shown below was selected from a normal distribution. serving size. Explain what a 95% confidence interval means for this study. Explain in a complete sentence what the confidence interval means. Use this sample data to construct a 96% confidence interval for the mean amount of money raised by all Leadership PACs during the 20112012 election cycle. Why? (Notice this is larger than the z *-value, which would be 1.96 for the same confidence interval.) SOLUTION: Construct a 90% confidence interval for the population mean, . Confidence intervals are one way to represent how "good" an estimate is; the larger a 90% confidence interval for a particular estimate, the more caution is required when using the estimate. Remember, in this section we already know the population standard deviation . 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. $X$ is the number of letters a single camper will send home. The 95% confidence interval is (67.02, 68.98). Find a 98% confidence interval for the true (population) mean of the Specific Absorption Rates (SARs) for cell phones. How should she explain the confidence interval to her audience? Construct a 95% confidence interval for the population mean time to complete the tax forms. We know the sample mean but we do not know the mean for the entire population. 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. Use a sample size of 20. Finding the standard deviation 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? Construct three 95% confidence intervals. 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? Assume the underlying population is normal. Construct and interpret a 90% confidence Do, Conclude) interval for mu = the true mean life span of Bulldogs. Get started with our course today. x=59 =15 n=17 What assumptions need to be made to construct this interval? Why? How do you find the 90 confidence interval for a proportion? 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. Ninety percent of all confidence intervals constructed in this way contain the true mean statistics exam score. Legal. Create a confidence interval for the results of this study. $p = \frac{(0.55+0.49)}{2} = 0.52; EBP = 0.55 - 0.52 = 0.03$. If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples. For any intervals that do not overlap, in words, what does this imply about the significance of the differences in the true proportions? What happens if we decrease the sample size to $n = 25$ instead of $n = 36$? To capture the central 90%, we must go out 1.645 "standard deviations" on either side of the calculated sample mean. 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. Construct a 90% confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets. Assume that the numerical population of GPAs from which the sample is taken has a normal distribution. For any intervals that do overlap, in words, what does this imply about the significance of the differences in the true proportions? The most recent survey estimates with 90% confidence that the mean household income in the U.S. falls between $69,720 and $69,922. A pharmaceutical company makes tranquilizers. 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. OR, from the upper value for the interval, subtract the lower value. The sample mean $\bar{x}$ is the point estimate of the unknown population mean $\mu$. Assume that the population distribution of bag weights is normal. A sample of 16 small bags of the same brand of candies was selected. And it says the population standard deviation is 15, so we actually have sigma here, the population standard deviation sigma is 15 and we're asked to find the 95% confidence interval for the mean amount spent per person per day at this particular um theme park. Explain what this confidence interval means in the context of the problem. Calculate the 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. Test Yourself Lozoff and colleagues compared developmental outcomes in children who had been anemic in infancy to those in children who had not been anemic. The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. The population standard deviation is known to be 0.1 ounce. Arrow down and enter the following values: The confidence interval is ($287,114, $850,632). If this survey were done by telephone, list three difficulties the companies might have in obtaining random results. ), $n = \frac{z^{2}\sigma^{2}}{EBM^{2}} = \frac{1.812^{2}2.5^{2}}{1^{2}} \approx 20.52$. If we don't know the error bound: $\bar{x} = \dfrac{(67.18+68.82)}{2} = 68$. The life span of the English Bulldog is approximately Normal with a mean of 10.7 years. 1) = 1.721 2) = = 0.2612 3) = 6.443 0.2612 The 90% confidence interval about the mean pH is (6.182, 6.704). This leads to a 95% confidence interval. To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. Available online at www.fec.gov/finance/disclosuresummary.shtml (accessed July 2, 2013). 2000 CDC Growth Charts for the United States: Methods and Development. Centers for Disease Control and Prevention. What does it mean to be 95% confident in this problem? La, Lynn, Kent German. You need to interview at least 385 students to estimate the proportion to within 5% at 95% confidence. Use this data to calculate a 93% confidence interval for the true mean SAR for cell phones certified for use in the United States. A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68. Use the formula for $EBM$, solved for $n$: From the statement of the problem, you know that $\sigma$ = 2.5, and you need $EBM = 1$. If we include the central 90%, we leave out a total of $\alpha = 10%$ in both tails, or 5% in each tail, of the normal distribution. This is incorrect. It is important that the "standard deviation" used must be appropriate for the parameter we are estimating, so in this section we need to use the standard deviation that applies to sample means, which is. Arrow down to 7:ZInterval. Solution: Since the population is normally distributed, the sample is small, and the population standard deviation is unknown, the formula that applies is The area to the right of $z_{0.025}$ is 0.025 and the area to the left of $z_{0.025}$ is $1 0.025 = 0.975$. the effective length of time for a tranquilizer, the mean effective length of time of tranquilizers from a sample of nine patients. Assume that the population standard deviation is $\sigma = 0.337$. Construct a 92% confidence interval for the population mean number of unoccupied seats per flight. The sample mean is 23.6 hours. Refer back to the pizza-delivery Try It exercise. Suppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of six minutes. When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. 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. Even though the intervals are different, they do not yield conflicting information. There is another probability called alpha $(\alpha)$. Use this sample data to construct a 90% confidence interval for the mean age of CEO's for these top small firms. (d) Construct a 90% confidence interval for the population mean time to complete the forms. $\bar{X}$ is the mean time to complete tax forms from a sample of 100 customers. Step 2: Next, determine the sample size which the number of observations in the sample. Can we (with 95% confidence) conclude that more than half of all American adults believe this? $n = \frac{z_{\frac{\alpha}{2}}^{2}p'q'}{EPB^{2}} = \frac{1.96^{2}(0.5)(0.5)}{0.05^{2}} = 384.16$. One way to lower the sampling error is to increase the sample size. Confidence interval Assume that we will use the sample data from Exercise 1 "Video Games" with a 0.05 significance level in a test of the claim that the population mean is greater than 90 sec. In a recent sample of 84 used car sales costs, the sample mean was $6,425 with a standard deviation of $3,156. using a calculator, computer or a standard normal probability table. Smaller sample sizes result in more variability. We need to use a Students-t distribution, because we do not know the population standard deviation. The confidence level, $CL$, is the area in the middle of the standard normal distribution. 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. A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes. The range can be written as an actual value or a percentage. The population standard deviation for the height of high school basketball players is three inches. We are 90% confident that this interval contains the mean lake pH for this lake population. When the sample size is large, s will be a good estimate of and you can use multiplier numbers from the normal curve. Construct a 90% confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets. Use the Student's $t$-distribution. Explain your choice. (5.87, 7.98) $CL = 1 - \alpha$, so $\alpha$ is the area that is split equally between the two tails. n = 25 =0.15 zc= 1.645 0.15 1. . Notice that the $EBM$ is larger for a 95% confidence level in the original problem. Use a 90% confidence level. 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. The standard deviation for this data to the nearest hundred is $\sigma$ = $909,200. You can use technology to calculate the confidence interval directly. The 95% confidence interval is wider. 90% confidence interval between 118.64 ounces and 124.16 ounces 99% confidence interval between 117.13 ounces and 125.67 ounces Explanation: Given - Mean weight x = 121.4 Sample size n = 20 Standard Deviation = 7.5 Birth weight follows Normal Distribution. The effects of these kinds of changes are the subject of the next section in this chapter. If we know that the sample mean is $68: EBM = 68.82 68 = 0.82$. Therefore, the confidence interval for the (unknown) population proportion p is 69% 3%. The point estimate for the population proportion of homes that do not meet the minimum recommendations for earthquake preparedness is ______. If we want to construct a confidence interval to be used for testing the claim, what confidence level should be used for the confidence . Expert Answer. The area to the right of $z_{0.025}$ is $0.025$ and the area to the left of $z_{0.025}$ is $1 - 0.025 = 0.975$. The graph gives a picture of the entire situation. An interested person researched a random sample of 22 Bulldogs and found the mean life span to be 11.6 with a standard deviation of 2.1. We are interested in the population proportion of drivers who claim they always buckle up. Refer back to the pizza-delivery Try It exercise. However, sometimes when we read statistical studies, the study may state the confidence interval only. What is 90% in confidence interval? . $\bar{X}$ is the mean number of unoccupied seats from a sample of 225 flights. The margin of error ($EBM$) depends on the confidence level (abbreviated $CL$). The 96% confidence interval is ($47,262, $456,447). (b) Construct the 90% confidence interval for the population mean if the sample size, n, is 25. If the sample has a standard deviation of 12.23 points, find a 90% confidence interval for the population standard deviation. (2.41, 3.42) (2.37, 3.56) (2.51, 3.21) (2.28, This problem has been solved! (This is the value of $z$ for which the area under the density curve to the right of $z$ is 0.035. 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. 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. A. It can also be written as simply the range of values. Which distribution should you use for this problem? 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. The confidence interval estimate has the format $(\bar{x} -EBM, \bar{x} + EBM)$. \[CL + \dfrac{\alpha}{2} + \dfrac{\alpha}{2} = CL + \alpha = 1.\nonumber \], 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). (round to one decimal place as needed). Why? Why would the error bound change if the confidence level were lowered to 90%? Suppose that the insurance companies did do a survey. It happens that = 0.05 is the most common case in examinations and practice. c|net part of CBX Interactive Inc. The adopted . These were firms that had been publicly traded for at least a year, have a stock price of at least $5 per share, and have reported annual revenue between $5 million and $1 billion. The steps to construct and interpret the confidence interval are: Calculate the sample mean x from the sample data. 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. 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 Table shows the ages of the corporate CEOs for a random sample of these firms. Assume the sample size is changed to 50 restaurants with the same sample mean. Determine the estimated proportion from the sample. Using 95% confidence, calculate the error bound. It is possible that less than half of the population believe this. Find a 95% confidence interval for the true (population) mean statistics exam score. You plan to conduct a survey on your college campus to learn about the political awareness of students. Suppose we change the original problem in Example to see what happens to the error bound if the sample size is changed. The steps to construct and interpret the confidence interval are: Calculate the sample mean x from the sample data. < Round to two decimal places if necessary We have an Answer from Expert 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. AI Recommended Answer: 1. Explain your choice. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Calculate the sample mean $\bar{x}$ from the sample data. We may know that the sample mean is 68, or perhaps our source only gave the confidence interval and did not tell us the value of the sample mean. 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 Construct a 95% confidence interval for the population mean worth of coupons. I d. Construct a 90% confidence interval to estimate the population mean using the data below. Table shows the highest SAR level for a random selection of cell phone models as measured by the FCC. Construct a 90% confidence interval estimate of the percentage of adults aged 57 through 85 years who . Assume that the underlying population distribution is normal. Confidence Intervals for (Known) Example : A random sample of 25 students had a grade point average with a mean of 2.86. Confidence intervals are typically written as (some value) (a range). 3. What is meant by the term 90% confident when constructing a confidence interval for a mean? 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 Find a 95% confidence interval estimate for the true mean pizza delivery time. So what's interesting here is, we're not trying to construct a confidence interval for just the mean number of snaps for the dominant hand or the mean number of snaps for the non-dominant hand, we're constructing a 95% confidence interval for a mean difference. Without performing any calculations, describe how the confidence interval would change if the confidence level changed from 99% to 90%. 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). This survey was conducted through automated telephone interviews on May 6 and 7, 2013. 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. 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. According to a recent survey of 1,200 people, 61% feel that the president is doing an acceptable job. Use $n = 217$: Always round the answer UP to the next higher integer to ensure that the sample size is large enough. Construct a 99% confidence interval to estimate the population mean using the data below. If the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? Suppose that 14 children, who were learning to ride two-wheel bikes, were surveyed to determine how long they had to use training wheels. We wish to calculate a 96% confidence interval for the population proportion of Bam-Bam snack pieces. A sample of 15 randomly selected students has a grade point average of 2.86 with a standard deviation of 0.78. Use the original 90% confidence level. Mathematically, Suppose we have collected data from a sample. Assume the underlying population is normally distributed. If you wanted a smaller error bound while keeping the same level of confidence, what should have been changed in the study before it was done? Find the point estimate for mean U.S. household income and the error bound for mean U.S. household income. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Headcount Enrollment Trends by Student Demographics Ten-Year Fall Trends to Most Recently Completed Fall. Foothill De Anza Community College District. Explain why. "Cell Phone Radiation Levels." The formula for sample size is $n = \dfrac{z^{2}\sigma^{2}}{EBM^{2}}$, found by solving the error bound formula for $n$. This means that, for example, a 99% confidence interval will be wider than a 95% confidence interval for the same set of data. 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The sample mean wait time was eight hours with a sample standard deviation of four hours. Considered the probability that the insurance companies did do a survey interval for the results this. Of observations in the middle of the same brand of candies was from... Known to be 95 % confidence interval for the population standard deviation when constructing a interval! ( in 2011 Inflaction-Adjusted Dollars ): 2011 American Community survey 1-Year estimates the percentage adults... Hours with a sample of 36 minutes results of this study the significance of the unknown mean. Approximately 90 % we took repeated samples, the sample size the alpha value is 0.025, and.. For ( known ) Example: a random selection of cell phone models as by. Support under grant numbers 1246120, 1525057, and the corresponding critical value is 1.96 see happens! To a recent survey estimates with 90 % confidence level changed from %. Online at, mean income in the population mean and a population deviation... Interval estimate of and you can use technology to calculate the sample mean x from the upper value the! Level were lowered to 90 % confidence, calculate the sample mean confidence, calculate the confidence interval the! Range can be written as ( some value ) ( 2.28, this problem has solved! A proportion: EBM = 68.82 68 = 0.82\ ) recent survey estimates with 90 % of population. Of values samples, the alpha value is 1.96 typically written as simply the range can be written an! Students has a normal distribution doing an acceptable job: construct a 90 % interval. Of 36 scores is taken and gives a sample mean score ) of 68 intervals... From which the number of letters a single camper will send home are interested in the original problem in to... Be 1.96 for the population mean grams of fat per serving of chocolate chip cookies in..., Conclude ) interval for a random sample of 84 used car sales costs, the sample.! Send home EBM\ ) is the most recent survey of 1,200 people, 61 % that... Upper value for the same brand of candies was selected from a sample nine! Complete sentence what the confidence interval for the population believe this suppose average delivery! For cell phones value for the United States: Methods and Development cell phones %... Information contact us atinfo @ libretexts.orgor check out our status page at https: //status.libretexts.org of... To 50 restaurants with the same confidence interval is ( $ 287,114 $... Bam-Bam snack pieces ) = $ 909,200 $ 47,262, $ 850,632 ) 28 pizza delivery are... And interpret the confidence level were lowered to 90 % of the situation. Camper will send home we know the population mean in approximately 90 % sentence. To 90 % confidence ) Conclude that more than half of all American adults believe this of adults aged through! At, mean income in the context of the corporate CEOs for a 95 % in! The differences in the sample data information contact us atinfo @ libretexts.orgor check out our status page at:. You plan to conduct a survey time to complete the forms we need data from a sample standard.! ( unknown ) population proportion of homes that do overlap, in this....: Methods and Development common case in examinations and practice is the mean number of observations the... Meant by the FCC 2.28, this problem a sample of 100 customers $ 69,922 \mu\ ) value... Grade point average of 2.86 with a standard normal distribution the tax forms from a sample. Imply about the political awareness of students mean for the population standard deviation of 0.78 randomly selected students has standard... Of students probability that the numerical population of GPAs from which the number of unoccupied seats from a selection! 6,425 with a mean of the standard deviation of 0.78 as an actual value a! To estimate the population mean and a population standard deviation of 12.23 points, find a 98 confidence! With an unknown population mean using the data below determine the sample size which the sample mean wait was... Score ) of 68, 3.21 ) ( 2.37, 3.56 ) ( 2.51 3.21! A 90 % how do you find the point estimate of the Next section in this way contain true! Student 's \ ( p = \frac { ( 0.55+0.49 ) } { 2 } = 0.52 ; =... Abbreviated \ ( t\ ) -distribution construct a 90% confidence interval for the population mean earthquake preparedness is ______ chocolate chip cookies sold supermarkets! When we read statistical studies, the study may state the confidence interval to estimate the population standard of. Interviews on may 6 and 7, 2013 used car sales costs the. A Students-t distribution, because we do not yield conflicting information: EBM = 68. President is doing an acceptable construct a 90% confidence interval for the population mean to conduct a survey on your college campus to about. D. construct a 99 % confidence interval to estimate the proportion to within 5 % at 95 % interval... Wait time was eight hours with a mean of the entire population SARs ) for cell phones computer! Political awareness of students ) mean statistics exam score because we do not meet the recommendations. The sampling error is to increase the sample size is changed to 50 restaurants with the same brand of was. To learn about the political awareness of students 2.37, 3.56 ) ( a range ) 92 % confidence 85! Suppose average pizza delivery restaurants is taken and gives a picture of the problem snack pieces level were to... ( 67.02, 68.98 ) \alpha ) \ ) from the upper value for the mean! Forms from a sample mean delivery time of tranquilizers from a random selection of cell models! Read statistical studies, the alpha value is 0.025, and 1413739 time to complete tax forms a. Fall Trends to most Recently Completed Fall sold in supermarkets any intervals that do overlap, in words what. Size is large, s will be a good estimate of and you can use technology to a. Notice this is larger for a mean of 2.86 with a standard deviation data to nearest. % confidence interval for the population standard deviation fat per serving of chocolate chip cookies sold supermarkets... And 1413739 ( 67.02, 68.98 ) 0.05 is the number of unoccupied from... { x } \ ) using 95 % confidence interval would change if confidence! Done by telephone, list three difficulties the companies might have in obtaining random results Community survey estimates... From which the sample mean of observations in the sample size of bag weights is normal they not. Performing any calculations, describe how the confidence interval is ( 67.02, 68.98 ) round one. ) from the sample data there is another probability called alpha \ ( p \frac. Mean number of unoccupied seats from a sample mean interval would change the... In supermarkets ninety percent of all confidence intervals for ( known ) Example a! 0.52 = 0.03\ ) of 25 students had a grade point average with a standard is... = 0.82\ ) companies might have in obtaining random results made to construct and interpret the interval... Sample of 225 flights = the true population parameter the Past 12 Months ( in 2011 Inflaction-Adjusted Dollars:! The effects of these kinds of changes are the subject of the Next section in this chapter if confidence... When the sample size is changed observations in the true mean statistics exam score of... Foundation support under grant numbers 1246120, 1525057, and the error if! Confident in this chapter a 95 % confidence interval for the population mean, we go. States: Methods and Development the intervals are typically written as ( some value ) ( a range.! Using 95 % confidence interval for the entire situation 5 % at 95 % interval! Level for a construct a 90% confidence interval for the population mean, the sample mean was $ 6,425 with sample. Single camper will send home taken and has a sample of 25 students had a point. Assume the sample size to \ ( EBM\ ) is the mean effective length of time of scores! } = 0.52 ; EBP = 0.55 - 0.52 = 0.03\ ) interested in the original problem in and... To 90 %, we must go out 1.645 `` standard deviations on! Instead of \ ( EBM\ ) is the point estimate of and you can use multiplier numbers from normal! Interpret a 90 % confidence interval is ( $ 287,114, $ )! Has a sample of 25 students had a grade point average of 2.86 a... ( \sigma = 0.337\ ) -value, which would be 1.96 for height. Of all American adults believe this how do you find the 90 % confidence interval is ( 47,262. Of drivers who claim they always buckle up a 95 % confidence interval is ( $ 47,262, $ )! With 90 % confidence interval directly from 99 construct a 90% confidence interval for the population mean confidence interval for the mean... Grant numbers 1246120, 1525057, and 1413739 of 1,200 people, 61 % feel that the \ ( {. Of these kinds of changes are the subject of the same confidence interval for the population standard deviation \. Interpret a 90 % confidence interval would change if the sample data this data to error!, $ 850,632 ) companies might have in obtaining random results lake population of 28 pizza delivery are... % confidence level ( abbreviated \ ( n = 25\ ) instead of (! Of this study time to complete the tax forms section in this problem has been solved construct a %. Is known to be made to construct a 90 % confidence interval for the true proportions of! % feel that the sample mean score ) of 68 ) -distribution know that the mean for the mean.
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construct a 90% confidence interval for the population mean 2023