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construct a 90% confidence interval 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. The 96% confidence interval is ($47,262, $456,447). Construct a 95% confidence interval for the population mean worth of coupons. The grams of fat per serving are as follows: 8; 8; 10; 7; 9; 9. Use a sample size of 20. It is interested in the mean amount of time individuals waste at the courthouse waiting to be called for jury duty. A survey of 20 campers is taken. A sample of 16 small bags of the same brand of candies was selected. 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 Solution: We first need to find the critical values: and. If we increase the sample size \(n\) to 100, we decrease the error bound. Interpret the confidence interval in the context of the problem. In words, define the random variable \(\bar{X}\). Construct a 90% confidence interval for the population mean weight of the candies. A national survey of 1,000 adults was conducted on May 13, 2013 by Rasmussen Reports. Construct a 90% confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets. How should she explain the confidence interval to her audience? Sketch the graph. There are 30 measures in the sample, so \(n = 30\), and \(df = 30 - 1 = 29\), \(CL = 0.96\), so \(\alpha = 1 - CL = 1 - 0.96 = 0.04\), \(\frac{\alpha}{2} = 0.02 t_{0.02} = t_{0.02} = 2.150\), \(EBM = t_{\frac{\alpha}{2}}\left(\frac{s}{\sqrt{n}}\right) = 2.150\left(\frac{521,130.41}{\sqrt{30}}\right) - $204,561.66\), \(\bar{x} - EBM = $251,854.23 - $204,561.66 = $47,292.57\), \(\bar{x} + EBM = $251,854.23+ $204,561.66 = $456,415.89\). We need to find the value of \(z\) that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution \(Z \sim N(0, 1)\). The stated \(\pm 3%\) represents the maximum error bound. The CONFIDENCE function calculates the confidence interval for the mean of the population. To find the confidence interval, start by finding the point estimate: the sample mean. \[EBM = \left(z_{\dfrac{\alpha}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right)\nonumber \], \[\alpha = 1 CL = 1 0.90 = 0.10\nonumber \], \[\dfrac{\alpha}{2} = 0.05 z_{\dfrac{\alpha}{2}} = z_{0.05}\nonumber \]. The sample size is less than 30. How would the number of people the firm surveys change? 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 \(\sigma\). If this survey were done by telephone, list three difficulties the companies might have in obtaining random results. 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 \(Z \sim N(0, 1)\). A point estimate for the true population proportion is: A 90% confidence interval for the population proportion is _______. The firm needs to determine what the confidence level should be, then apply the error bound formula to determine the necessary sample size. 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. \(CL = 0.95 \alpha = 1 - 0.95 = 0.05 \frac{\alpha}{2} = 0.025 z_{\frac{\alpha}{2}} = 1.96.\) Use \(p = q = 0.5\). An example of how to calculate a confidence interval for a mean. 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. Notice that the \(EBM\) is larger for a 95% confidence level in the original problem. These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax. Create a 99% confidence interval for the true proportion of American adults who have illegally downloaded music. 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. \[z_{\dfrac{\alpha}{2}} = z_{0.05} = 1.645\nonumber \]. We will use a Students \(t\)-distribution, because we do not know the population standard deviation. OR, from the upper value for the interval, subtract the lower value. Assuming a population standard deviation of 0.2 mph, construct a 90% confidence interval for the mean difference between true speed and indicated speed for all vehicles. What happens if we decrease the sample size to \(n = 25\) instead of \(n = 36\)? The range can be written as an actual value or a percentage. The reason that we would even want to create, How to Perform Logistic Regression in Excel, How to Perform a Chi-Square Goodness of Fit Test in Excel. The 98% confidence interval of the population mean amount of mercury in tuna sushi is equal to (0.287 ppm, 1.151 ppm) . Which distribution should you use for this problem? Required fields are marked *. State the confidence interval. What assumptions need to be made to construct this interval? . \(N 7.9\left(\frac{2.5}{\sqrt{20}}\right)\). We need to use a Students-t distribution, because we do not know the population standard deviation. This means that there is a 95% probability the population mean would fall within the confidence interval range 95 is not a standard significance value for confidence. Find the point estimate for mean U.S. household income and the error bound for mean U.S. household income. In six packages of The Flintstones Real Fruit Snacks there were five Bam-Bam snack pieces. Explain what a 97% confidence interval means for this study. (Round to 2 decimal places) 0.26 (e) If the Census did another survey, kept the error bound the same, and surveyed only 50 people instead of 200, what would happen to the level of confidence? The population is skewed to one side. You can use technology to calculate the confidence interval directly. Construct a 90% confidence interval for the population proportion of people who feel the president is doing an acceptable job. Recall, when all factors remain unchanged, an increase in sample size decreases variability. Suppose we change the original problem in Example by using a 95% confidence level. 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. The sample mean is 23.6 hours. Construct a 92% confidence interval for the population mean number of unoccupied seats per flight. Even though the three point estimates are different, do any of the confidence intervals overlap? With a 90 percent confidence interval, you have a 10 percent chance of being wrong. 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. Even though the intervals are different, they do not yield conflicting information. The 90% confidence interval is (67.1775, 68.8225). During the 2012 campaign season, there were 1,619 candidates for the House of Representatives across the United States who received contributions from individuals. The following data were collected: 20; 75; 50; 65; 30; 55; 40; 40; 30; 55; $1.50; 40; 65; 40. Stanford University conducted a study of whether running is healthy for men and women over age 50. Aconfidence interval for a meanis a range of values that is likely to contain a population mean with a certain level of confidence. In Exercises 9-24, construct the confidence interval estimate of the mean. We wish to calculate a 96% confidence interval for the population proportion of Bam-Bam snack pieces. Find the point estimate for the population mean. 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. 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. Find the point estimate and the error bound for this confidence interval. We are 90% confident that this interval contains the mean lake pH for this lake population. Use the Student's t-distribution. 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. We wish to construct a 95% confidence interval for the mean height of male Swedes. How to interpret a confidence interval for a mean. 9.1 - Confidence Intervals for a Population Proportion A random sample is gathered to estimate the percentage of American adults who believe that parents should be required to vaccinate their children for diseases like measles, mumps, and rubella. Decreasing the sample size causes the error bound to increase, making the confidence interval wider. Explain in a complete sentence what the confidence interval means. A camp director is interested in the mean number of letters each child sends during his or her camp session. Find the confidence interval at the 90% Confidence Level for the true population proportion of southern California community homes meeting at least the minimum recommendations for earthquake preparedness. Short Answer. This is the t*- value for a 95 percent confidence interval for the mean with a sample size of 10. 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. Past studies have shown that the standard deviation is 0.15 and the population is normally distributed. Notice the difference in the confidence intervals calculated in Example and the following Try It exercise. Use the following information to answer the next two exercises: Five hundred and eleven (511) homes in a certain southern California community are randomly surveyed to determine if they meet minimal earthquake preparedness recommendations. It happens that = 0.05 is the most common case in examinations and practice. The confidence level for this study was reported at 95% with a \(\pm 3%\) margin of error. A 98% confidence interval for mean is [{Blank}] . Available online at www.fec.gov/data/index.jsp (accessed July 2, 2013). 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. The area to the right of \(z_{0.05}\) is \(0.05\) and the area to the left of \(z_{0.05}\) is \(1 - 0.05 = 0.95\). 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. Explain what a 95% confidence interval means for this study. Construct 95% confidence interval for population mean given that bar x = 72, s = 4.8, n = 36. 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 many students must you interview? Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 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. This is 345. The life span of the English Bulldog is approximately Normal with a mean of 10.7 years. Some people think this means there is a 90% chance that the population mean falls between 100 and 200. Construct a 95% confidence interval for the population mean length of time. If many random samples were taken of size 14, what percent of the confidence intervals constructed should contain the population mean worth of coupons? 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? Arrow to Stats and press ENTER. From the problem, we know that \(\sigma = 15\) and \(EBM = 2\). However, sometimes when we read statistical studies, the study may state the confidence interval only. Create a 95% confidence interval for the mean total individual contributions. Define the random variable \(\bar{X}\) in words. It is assumed that the distribution for the length of time they last is approximately normal. The population standard deviation for the height of high school basketball players is three inches. Use the Student's t-distribution. Increasing the confidence level increases the error bound, making the confidence interval wider. Is the mean within the interval you calculated in part a? 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. What value of 2* should be used to construct a 95% confidence interval of a population mean? The point estimate for the population proportion of homes that do not meet the minimum recommendations for earthquake preparedness is ______. The total number of snack pieces in the six bags was 68. When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. Step 1: Identify the sample mean {eq}\bar {x} {/eq}, the sample size {eq}n {/eq}, and the sample standard. (Round to two decimal places as needed.) Sample Variance Construct a 99% confidence interval to estimate the population mean using the data below. 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 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. What is the error bound? Suppose we know that a confidence interval is (67.18, 68.82) and we want to find the error bound. \(n = \frac{z_{\frac{\alpha}{2}}^{2}p'q'}{EPB^{2}} = \frac{1.96^{2}(0.5)(0.5)}{0.05^{2}} = 384.16\). The following table shows the total receipts during this cycle for a random selection of 20 Leadership PACs. \(\bar{X}\) is normally distributed, that is, \(\bar{X} \sim N(\mu_{x},\dfrac{\sigma}{\sqrt{n}})\). If we don't know the sample mean: \(EBM = \dfrac{(68.8267.18)}{2} = 0.82\). Construct a 90 % confidence interval to estimate the population mean using the accompanying data. Did you expect it to be? 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. This means Assume the underlying population is normally distributed. Get started with our course today. If we want to be 95% confident that the sample mean height is within one inch of the true population mean height, how many randomly selected students must be surveyed? From the upper value for the interval, subtract the sample mean. Find the 95% Confidence Interval for the true population mean for the amount of soda served. If we know that the sample mean is \(68: EBM = 68.82 68 = 0.82\). Different phone models have different SAR measures. The sample mean \(\bar{x}\) is the point estimate of the unknown population mean \(\mu\). Use this sample data to construct a 90% confidence interval for the mean age of CEOs for these top small firms. An icon used to represent a menu that can be toggled by interacting with this icon. Explain your choice. What will happen to the error bound and confidence interval if 500 community colleges were surveyed? The formula to create a confidence interval for a mean. Why would the error bound change if the confidence level were lowered to 90%? document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Statology is a site that makes learning statistics easy by explaining topics in simple and straightforward ways. Legal. \(P =\) the proportion of people in a sample who feel that the president is doing an acceptable job. 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. 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. \(CL = 0.75\), so \(\alpha = 1 0.75 = 0.25\) and \(\frac{\alpha}{2} = 0.125 z_{\frac{\alpha}{2}} = 1.150\). Construct a 95% confidence interval for the population mean length of engineering conferences. It will need to change the sample size. Yes, the intervals (0.72, 0.82) and (0.65, 0.76) overlap, and the intervals (0.65, 0.76) and (0.60, 0.72) overlap. Explain why. This means that those doing the study are reporting a maximum error of 3%. You know that the average length is 7.5 inches, the sample standard deviation is 2.3 inches, and the sample size is 10. Remember, in this section we already know the population standard deviation . The sample standard deviation is 2.8 inches. That's a lot. \(\sigma = 3; n = 36\); The confidence level is 95% (CL = 0.95). 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. When the sample size is large, s will be a good estimate of and you can use multiplier numbers from the normal curve. The margin of error (\(EBM\)) depends on the confidence level (abbreviated \(CL\)). \(\bar{X}\) is the mean number of letters sent home from a sample of 20 campers. The 95% confidence interval is (67.02, 68.98). "Cell Phone Radiation Levels." \(N\left(23.6, \frac{7}{\sqrt{100}}\right)\) because we know sigma. Construct a 95% confidence interval for the population mean time wasted. This means that we can proceed with finding a 95% confidence interval for the population variance. Construct a 99% confidence interval for the population mean length of time using training wheels. Consequently, P{' 1 (X) < < ' 2 (X)} = 0.95 specifies {' 1 (X), ' 2 (X)} as a 95% confidence interval for . 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. We are interested in the population proportion of people who feel the president is doing an acceptable job. 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. The confidence interval is (to three decimal places)(67.178, 68.822). 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. As for the population of students in the MRPA, it represents 12%. Define the random variables \(X\) and \(P\) in words. The population standard deviation is six minutes and the sample mean deliver time is 36 minutes. (2.41, 3.42) (2.37, 3.56) (2.51, 3.21) (2.28, This problem has been solved! Refer back to the pizza-delivery Try It exercise. Find a 90% confidence interval for the true (population) mean of statistics exam scores. 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. \(\bar{X}\) is the mean time to complete tax forms from a sample of 100 customers. Arrow down to Calculate and press ENTER. 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. { "7.01:_Confidence_Intervals" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.
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