Construct a 90% confidence interval for the population proportion of people who feel the president is doing an acceptable job. The population is skewed to one side. 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%). ), \(EBM = (1.96)\left(\dfrac{3}{\sqrt{36}}\right) = 0.98\). Statology Study is the ultimate online statistics study guide that helps you study and practice all of the core concepts taught in any elementary statistics course and makes your life so much easier as a student. The adopted . The confidence interval is expressed as a percentage (the most frequently quoted percentages are 90%, 95%, and 99%). The mean weight was two ounces with a standard deviation of 0.12 ounces. \(\bar{X}\) is the mean time to complete tax forms from a sample of 100 customers. The standard deviation for this data to the nearest hundred is \(\sigma\) = $909,200. 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. \(X\) is the number of unoccupied seats on a single flight. As previously, assume that the population standard deviation is \(\sigma = 0.337\). Thus, we do not need as large an interval to capture the true population mean. Arrow to Stats and press ENTER. You plan to conduct a survey on your college campus to learn about the political awareness of students. However, sometimes when we read statistical studies, the study may state the confidence interval only. Construct a 92% confidence interval for the population mean number of unoccupied seats per flight. A point estimate for the true population proportion is: A 90% confidence interval for the population proportion is _______. Which? For example, if we constructed 100 of these confidence intervals, we would expect 90 of them to contain the true population mean exam score. Construct a 98% confidence interval for the population mean weight of the candies. This is the t*- value for a 95 percent confidence interval for the mean with a sample size of 10. The population standard deviation is six minutes and the sample mean deliver time is 36 minutes. \[z_{\dfrac{\alpha}{2}} = z_{0.025} = 1.96\nonumber \]. \(z_{\dfrac{\alpha}{2}} = z_{0.025} = 1.96\), when using invnorm(0.975,0,1) on the TI-83, 83+, or 84+ calculators. Mathematically, Suppose we have collected data from a sample. The CONFIDENCE function calculates the confidence interval for the mean of the population. In words, define the random variable \(X\). \(EBM = (z_{0.01})\dfrac{\sigma}{\sqrt{n}} = (2.326)\dfrac{0.337}{\sqrt{30}} =0.1431\). 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. How to interpret a confidence interval for a mean. Assume the underlying population is normal. View A7DBAEA8-E1D4-4235-90E6-13F3575EA3F9.jpeg from STATISTICS 1001 at Western Governors University. The FEC has reported financial information for 556 Leadership PACs that operating during the 20112012 election cycle. Statistics Statistical Inference Overview Confidence Intervals 1 Answer VSH Feb 22, 2018 Answer link We know the standard deviation for the population, and the sample size is greater than 30. A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes. A survey of 20 campers is taken. Construct a 90% confidence interval for the population mean weight of the candies. If we know that the sample mean is \(68: EBM = 68.82 68 = 0.82\). 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. Do you think that six packages of fruit snacks yield enough data to give accurate results? Suppose a large airline wants to estimate its mean number of unoccupied seats per flight over the past year. (a) Construct the 90% confidence interval for the population mean if the sample size, n, is 15. A 90% confidence interval for a population mean is determined to be 800 to 900. That's a lot. 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 \(\sigma\). Notice that the \(EBM\) is larger for a 95% confidence level in the original problem. State the confidence interval. 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. Explain what a 95% confidence interval means for this study. The mean from the sample is 7.9 with a sample standard deviation of 2.8. 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. Given that the population follows a normal distribution, construct a 90% confidence interval estimate of the mean of the population. 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. It is assumed that the distribution for the length of time they last is approximately normal. Available online at www.cdc.gov/growthcharts/2000thchart-us.pdf (accessed July 2, 2013). Why? A sample of 16 small bags of the same brand of candies was selected. We wish to construct a 95% confidence interval for the mean height of male Swedes. A political action committee (PAC) is a committee formed to raise money for candidates and campaigns. The sample size is less than 30. If we increase the sample size \(n\) to 100, we decrease the error bound. \(\alpha\) is related to the confidence level, \(CL\). 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. 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. \(P =\) the proportion of people in a sample who feel that the president is doing an acceptable job. One hundred seventy-three (173) of the homes surveyed met the minimum recommendations for earthquake preparedness, and 338 did not. 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. The percentage reflects the confidence level. Decreasing the sample size causes the error bound to increase, making the confidence interval wider. To capture the central 90%, we must go out 1.645 "standard deviations" on either side of the calculated sample mean. Confidence Intervals for (Known) Example : A random sample of 25 students had a grade point average with a mean of 2.86. 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. 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. Use the following information to answer the next three exercises: According to a Field Poll, 79% of California adults (actual results are 400 out of 506 surveyed) feel that education and our schools is one of the top issues facing California. Suppose that our sample has a mean of \(\bar{x} = 10\) and we have constructed the 90% confidence interval (5, 15) where \(EBM = 5\). percent of all Asians who would welcome a black person into their families. A pharmaceutical company makes tranquilizers. That means that tn - 1 = 1.70. Since we increase the confidence level, we need to increase either our error bound or the sample size. It is possible that less than half of the population believe this. 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. The error bound and confidence interval will decrease. That is, theres only a 5% chance that the true population mean weight of turtles is greater than 307.25 pounds or less than 292.75 pounds. Assume that the numerical population of GPAs from which the sample is taken has a normal distribution. \(N 7.9\left(\frac{2.5}{\sqrt{20}}\right)\). Suppose we collect a random sample of turtles with the following information: Here is how to find various confidence intervals for the true population mean weight: 90% Confidence Interval:300 +/- 1.645*(18.5/25) =[293.91, 306.09], 95% Confidence Interval:300 +/- 1.96*(18.5/25) =[292.75, 307.25], 99% Confidence Interval:300 +/- 2.58*(18.5/25) = [290.47,309.53]. 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. If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples. What value of 2* should be used to construct a 95% confidence interval of a population mean? n = 25 =0.15 zc= 1.645 0.15 1. . x=59 =15 n=17 What assumptions need to be made to construct this interval? Construct a 90% confidence interval to estimate the population mean using the data below. 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. Use the following information to answer the next two exercises: A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16 oz. These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax. For 36 vehicles tested the mean difference was $-1.2$ mph. Updated 2021 - https://youtu.be/Ob0IulZFU6sIn this video I show you how to use statcrunch to quickly create a Confidence Interval for a Population Mean. Remember, in this section we already know the population standard deviation \(\sigma\). Another question in the poll was [How much are] you worried about the quality of education in our schools? Sixty-three percent responded a lot. 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. (Explain what the confidence interval means, in the words of the problem.). How would you interpret this statement? Suppose we want to lower the sampling error. The following table shows the z-value that corresponds to popular confidence level choices: Notice that higher confidence levels correspond to larger z-values, which leads to wider confidence intervals. Some of the data are shown in the table below. A. It concluded with 95% confidence that 49% to 55% of Americans believe that big-time college sports programs corrupt the process of higher education. Get started with our course today. Six different national brands of chocolate chip cookies were randomly selected at the supermarket. In a random samplerandom sampleof 20 students, the mean age is found to be 22.9 years. What is one way to accomplish that? 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. The area to the right of \(z_{0.05}\) is \(0.05\) and the area to the left of \(z_{0.05}\) is \(1 - 0.05 = 0.95\). For a two-tailed 95% confidence interval, the alpha value is 0.025, and the corresponding critical value is 1.96. Arrow down and enter the following values: The confidence interval is (to three decimal places) (0.881, 1.167). Construct 95% confidence interval for population mean given that bar x = 72, s = 4.8, n = 36. Press ENTER. 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. In words, define the random variable \(\bar{X}\). Define the random variables \(X\) and \(P\), in words. Go to the store and record the grams of fat per serving of six brands of chocolate chip cookies. 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. The point estimate for the population proportion of homes that do not meet the minimum recommendations for earthquake preparedness is ______. How do you find the 90 confidence interval for a proportion? Subtract the error bound from the upper value of the confidence interval. 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. Use this sample data to construct a 90% confidence interval for the mean age of CEOs for these top small firms. Without performing any calculations, describe how the confidence interval would change if the confidence level changed from 99% to 90%. 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. \(\alpha\) is the probability that the interval does not contain the unknown population parameter. Why or why not? Assume the underlying distribution is approximately normal. As for the population of students in the MRPA, it represents 12%. The steps to construct and interpret the confidence interval are: Calculate the sample mean x from the sample data. Assume that the population distribution of bag weights is normal. In Equation \ref{samplesize}, \(z\) is \(z_{\dfrac{a}{2}}\), corresponding to the desired confidence level. The stated \(\pm 3%\) represents the maximum error bound. Suppose we have data from a sample. What does it mean to be 95% confident in this problem? Then divide the difference by two. Define the random variables \(X\) and \(P\), in words. An article regarding interracial dating and marriage recently appeared in the Washington Post. 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? 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. Past studies have shown that the standard deviation is 0.15 and the population is normally distributed. Assume the population has a normal distribution. The population standard deviation for the height of high school basketball players is three inches. I d. Create a 99% confidence interval for the true proportion of American adults who have illegally downloaded music. Why or why not? Find the 95% Confidence Interval for the true population mean for the amount of soda served. OR, from the upper value for the interval, subtract the lower value. The population standard deviation for the age of Foothill College students is 15 years. To capture the true population mean, we need to have a larger interval. 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. When \(n = 25: EBM = \left(z_{\dfrac{\alpha}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right) = (1.645)\left(\dfrac{3}{\sqrt{25}}\right) = 0.987\). This is incorrect. \(\bar{X}\) is the mean number of letters sent home from a sample of 20 campers. Find a 90% confidence interval for the true (population) mean of statistics exam scores. Solution: Since the population is normally distributed, the sample is small, and the population standard deviation is unknown, the formula that applies is Construct a 90% confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets. It is denoted by. The sample mean is 13.30 with a sample standard deviation of 1.55. The mean length of the conferences was 3.94 days, with a standard deviation of 1.28 days. 2000 CDC Growth Charts for the United States: Methods and Development. Centers for Disease Control and Prevention. If we took repeated samples, approximately 90% of the samples would produce the same confidence interval. The mean delivery time is 36 minutes and the population standard deviation is six minutes. \(\bar{X}\) is normally distributed, that is, \(\bar{X} \sim N(\mu_{x},\dfrac{\sigma}{\sqrt{n}})\). In this survey, 86% of blacks said that they would welcome a white person into their families. Assume the sample size is changed to 50 restaurants with the same sample mean. It is denoted by n. The sample standard deviation is 2.8 inches. (The area to the right of this \(z\) is 0.125, so the area to the left is \(1 0.125 = 0.875\).). Legal. Confidence Interval Calculator for the Population Mean. 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. \(X =\) 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. Thus, a 95% confidence interval for the true daily discretionary spending would be $ 95 2 ( $ 4.78) or $ 95 $ 9.56. What is 90% in confidence interval? The population distribution is assumed to be normal. The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is \(\pm 3%\). No, the confidence interval includes values less than or equal to 0.50. A sample of 15 randomly selected students has a grade point average of 2.86 with a standard deviation of 0.78. Determine the estimated proportion from the sample. To find the 98% confidence interval, find \(\bar{x} \pm EBM\). Assume the population has a normal distribution. \(CL = 0.75\), so \(\alpha = 1 0.75 = 0.25\) and \(\frac{\alpha}{2} = 0.125 z_{\frac{\alpha}{2}} = 1.150\). What is the confidence interval estimate for the population mean? The confidence level, \(CL\), is the area in the middle of the standard normal distribution. Note:You can also find these confidence intervals by using the Statology Confidence Interval Calculator. We will use a Students \(t\)-distribution, because we do not know the population standard deviation. The population standard deviation is known to be 2.5. 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. We are interested in the population proportion of people who feel the president is doing an acceptable job. OR, average the upper and lower endpoints of the confidence interval. The way we would interpret a confidence interval is as follows: There is a 95% chance that the confidence interval of [292.75, 307.25] contains the true population mean weight of turtles. What is meant by the term 90% confident when constructing a confidence interval for a mean? { "7.01:_Confidence_Intervals" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.
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