The population standard deviation is known to be 2.5. (f) An interval that is calculated so that it has known likelihood of capturing the parameter. X is the height of a Swedish male, and is the mean height from a sample of 48 Swedish males. The LibreTexts libraries are Powered by MindTouch® and 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. Assume the underlying distribution is approximately normal. Construct a 95% confidence interval for the population mean height of male Swedes. Solution: A. Determine the estimated proportion from the sample. 5th ISTA seminar on statistics S Grégoire August 1999 Confidence interval Confidence interval Exercises: 1. Practice Problems: Confidence Intervals. The grams of fat per serving are as follows: 8; 8; 10; 7; 9; 9. Six different national brands of chocolate chip cookies were randomly selected at the supermarket. 11. Why? Chapter 8, Exercise Solutions, Principles of Econometrics, 3e 183 EXERCISE 8.5 (a) The table below displays the 95% confidence intervals obtained using the critical t-value t(0.975,497) =1.965 and both the least squares standard errors and the White’s standard errors. Define the random variables \(X\) and \(P′\), in words. ), \(n = \frac{z^{2}\sigma^{2}}{EBM^{2}} = \frac{1.812^{2}2.5^{2}}{1^{2}} \approx 20.52\). Exercise 1: Confidence Interval A machine is set up such that the average content of juice per bottle equals µ. Legal. We estimate with 96% confidence that the mean amount of money raised by all Leadership PACs during the 2011–2012 election cycle lies between $47,292.57 and $456,415.89. In one to three complete sentences, explain what the ±3% represents. \(X =\) 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. Construct a 96% confidence interval for the population proportion of Bam-Bam snack pieces per bag. M1 = {22, 24}, M1 = {22, 26}, M1 = {24, 26}. Finding Critical Values and 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? Compare this to the acceptance region Knowing this, determine the sample size for each stratum. \(N 7.9\left(\frac{2.5}{\sqrt{20}}\right)\). Assuming that the usage is normally distributed, provide an expression for calculating a 99% confidence interval for the mean usage in the March quarter of 2006. Use the formula for \(EBM\), solved for \(n\): From the statement of the problem, you know that \(\sigma\) = 2.5, and you need \(EBM = 1\). Why or why not? 1.Write down all possible samples of size two, chosen by simple random sampling. Why would the error bound change if the confidence level were lowered to 90%? Define the random variables \(X\) and \(P′\) in words. 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. Researchers in a hospital used the drug on a random sample of nine patients. The average height of young adult males has a normal distribution with standard deviation of 2.5 inches. Assume the underlying population is normal. (The area to the right of this \(z\) is 0.125, so the area to the left is \(1 – 0.125 = 0.875\).). percent of all Asians who would welcome a white person into their families. A statistical study of sales in the last nine months has found a confidence interval for the mean of monthly sales with extremes of $4,663 and $5,839. Have questions or comments? An article regarding interracial dating and marriage recently appeared in the Washington Post. Explain your choice. What is the meaning of a confidence interval ? Suppose that a committee is studying whether or not there is waste of time in our judicial system. The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is \(\pm 3%\). All that we would need to do is to take square roots of the endpoints. Public Policy Polling recently conducted a survey asking adults across the U.S. about music preferences. Prerequisites. 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 the Student's t-distribution. Consequently, we prefer to make sure that any words that imply probability or variability are clearly attached to the words 'confidence interval… Even though the three point estimates are different, do any of the confidence intervals overlap? \(n = \frac{z_{\frac{\alpha}{2}}^{2}p'q'}{EPB^{2}} = \frac{1.96^{2}(0.5)(0.5)}{0.05^{2}} = 384.16\). Note that we are not given the population standard deviation, only the standard deviation of the sample. From the small data set in 2006, the sample standard deviation is 91.5kWh. Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. Can we (with 95% confidence) conclude that more than half of all American adults believe this? Create a confidence interval for the results of this study. One of the questions asked was “What is the main problem facing the country?” Twenty percent answered “crime.” We are interested in the population proportion of adult Americans who feel that crime is the main problem. 3.Calculate the variance of the sample averages. Arrow down and enter the name of the list where the data is stored. Assume the underlying population is normally distributed. I like to spend my time reading, gardening, running, learning languages and exploring new places. According to the error bound formula, the firm needs to survey 206 people. 1.In a sample of 400 selected at random, a sample mean of 50 was obtained. Note this is a probability statement about the confidence interval, not the population parameter. Since we increase the confidence level, we need to increase either our error bound or the sample size. Without performing any calculations, describe how the confidence interval would change if the confidence level changed from 99% to 90%. A point estimate for the true population proportion is: A 90% confidence interval for the population proportion is _______. Regarding it as a random sample of all high school students, use it to construct a \(99\%\) confidence interval for the mean SAT score of all students. How many male students must you measure? The committee randomly surveyed 81 people who recently served as jurors. In Exercises 17, construct the confidence interval for? We will use a Student’s \(t\)-distribution, because we do not know the population standard deviation. t Distribution: Computes areas of the t distribution. Construct a 95% confidence interval for the population proportion who claim they always buckle up. 13. This survey was conducted through automated telephone interviews on May 6 and 7, 2013. Large Data Set Exercises. A pharmaceutical company makes tranquilizers. Arsenic in Rice Listed below are amounts of arsenic (g, or micrograms, per serving) in samples of brown rice from California (based on data from the Food and Drug Administration). Explain any differences between the values. Explain your choice. Explain why. Explain in a complete sentence what the confidence interval means. Arrow down and enter the following values: The confidence interval is ($287,114, $850,632). 1. Explain what this confidence interval means in the context of the problem. Number of minutes spent is Stage IV sleep is recorded for sixty-one patients. slightly different 95% confidence interval to the first experiment. We are interested in the population proportion of people who feel the president is doing an acceptable job. Assume that the underlying population distribution is normal. Confidence Interval Exercises. The average heights of a random sample of 400 people from a city is 1.75 m. It is known that the heights of the population are random variables that follow a normal distribution with a variance of 0.16 m. The sample must be at least 1,083 people. It's simple: each one of our tutorial videos explains how to answer one of the exam questions provided. The mean from the sample is 7.9 with a sample standard deviation of 2.8. Determine a 95% confidence interval for the difference between the mean heights of cross-fertilized and self-fertilized Zea mays. \(CL = 0.95 \alpha = 1 - 0.95 = 0.05 z_{\frac{\alpha}{2}} = 1.96\). If convenient, use technology to construct the confidence interval. A political action committee (PAC) is a committee formed to raise money for candidates and campaigns. \(\bar{X}\) is the mean number of unoccupied seats from a sample of 225 flights. Is the mean within the interval you calculated in part a? We need to use a Student’s-t distribution, because we do not know the population standard deviation. Determine the confidence interval with a confidence level of 97% for the average population. Which distribution should you use for this problem? What percentage of (1-alpha)100% confidence intervals for µ will fail to capture µ? It concluded with 95% confidence that 49% to 55% of Americans believe that big-time college sports programs corrupt the process of higher education. Construct a 95% confidence interval for the population mean length of engineering conferences. A Leadership PAC is a PAC formed by a federal politician (senator or representative) to raise money to help other candidates’ campaigns. In Exercises 5–8, use the given information to find the number of degrees of freedom, the critical values χ 2 L and X 2 R, and the confidence interval estimate of σ.The samples are from Appendix B and it is reasonable to assume that a simple random sample has been selected from a population with a normal distribution. When inferring statistical values based upon a sample, there is a band of uncertainty around the sample statistic in which the population statistic lies. Therefore, the researchers decide to choose the previous sample using stratified sampling as it is known that the recreations of the inhabitants change with age. Why? 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. Construct a 92% confidence interval for the population mean number of unoccupied seats per flight. Therefore, the researchers decide to choose the previous sample using stratified … In addition to a confidence interval, find a one-sided confidence bound and interpret. For any intervals that do overlap, in words, what does this imply about the significance of the differences in the true proportions? 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. Given the population of elements: {22, 24, 26}. \(N\left(0.61, \sqrt{\frac{(0.61)(0.39)}{1200}}\right)\). yˆ i yˆ i #2 (a) Write the new regression model. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. 100 individuals are selected at random and surveyed. 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. What were the average sales over the nine month period? It is known that 2,500 children, 7,000 adults and 500 elderly live in the neighborhood. (This is the value of \(z\) for which the area under the density curve to the right of \(z\) is 0.035. Why? Create a 99% confidence interval for the true proportion of American adults who have illegally downloaded music. If we were to sample many groups of nine patients, 95% of the samples would contain the true population mean length of time. What will happen to the error bound and confidence interval if 500 campers are surveyed? In a department store chain, 150 people work in human resources, 450 in sales, 200 in accounting and 100 in customer service. The following table shows the total receipts during this cycle for a random selection of 20 Leadership PACs. (5 marks) The volume, X millilitres, of hand cream in tubs is normally distributed with mean and variance r; . Thus, they estimate the percentage of adult Americans who feel that crime is the main problem to be between 18% and 22%. Define the random variable \(X\) in words. (b) What change in gasoline mileage is associated with a 1 cm3 change is engine displacement? Assume the underlying distribution is approximately normal. We are interested in the population proportion of drivers who claim they always buckle up. The firm needs to determine what the confidence level should be, then apply the error bound formula to determine the necessary sample size. \(X\) is the number of letters a single camper will send home. \(\bar{X}\) is the mean number of letters sent home from a sample of 20 campers. Construct a 90% confidence interval for the population mean number of letters campers send home. The height of students studying at a language school follows a normal distribution with a mean of 1.62 m and a standard deviation of 0.12. 2.With a confidence level of 90%, what would the minimum sample size need to be in order for the true mean of the heights to be less than 2 cm from the sample mean? Go to the store and record the grams of fat per serving of six brands of chocolate chip cookies. percent of all Asians who would welcome a Latino into their families. e. The confidence interval will decrease in size, because the sample size increased. 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. fail. The result would be a 95% confidence interval for the standard deviation. Forty-eight male Swedes are surveyed. The proportion of color blind individuals in a population needs to be estimated by the percentage observed in a random sample of individuals of size n. 1.If the percentage of color blind individuals in the sample is 30%, estimate the value of n so that, with a confidence level of 0.95, the error in the estimate is less than 3.1%. The 96% confidence interval is ($47,262, $456,447). Table shows the total receipts from individuals for a random selection of 40 House candidates rounded to the nearest $100. Exercises 9.2. A sample of the various prices for a particular product has been conducted in 16 stores which were selected at random in a neighborhood of a city. This means that those doing the study are reporting a maximum error of 3%. The American Community Survey (ACS), part of the United States Census Bureau, conducts a yearly census similar to the one taken every ten years, but with a smaller percentage of participants. Announcements for 84 upcoming engineering conferences were randomly picked from a stack of IEEE Spectrum magazines. Construct a 95% confidence Interval for 19, giving the limits to the nearest integer. Author(s) David M. Lane. The full step-by-step solution to problem: 21BSC from chapter: 7.3 was answered by , our top Statistics solution expert on 03/15/17, 10:30PM. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. The most recent survey estimates with 90% confidence that the mean household income in the U.S. falls between $69,720 and $69,922. The following data were collected: 20¢; 75¢; 50¢; 65¢; 30¢; 55¢; 40¢; 40¢; 30¢; 55¢; $1.50; 40¢; 65¢; 40¢. 100 individuals are selected at random and surveyed. It was revealed that they used them an average of six months with a sample standard deviation of three months. We are interested in the population proportion of adult Americans who are worried a lot about the quality of education in our schools. They randomly surveyed 400 drivers and found that 320 claimed they always buckle up. In order to conduct a survey, a sample of 180 workers is selected. The total number of snack pieces in the six bags was 68. How would the number of people the firm surveys change? Of course, since the standard deviation is the square root of the variance, this method could be used to construct a confidence interval for the population standard deviation. What is the probability that the mean of a random sample of 100 students will be taller than 1.60 m? Find the point estimate and the error bound for this confidence interval. Assume a population standard deviation of $200. Construct a 95% confidence interval for the population mean length of time. The 95% confidence interval for β2 is given by bt b22±=±× =−cse( ) 0.0761 2.074 0.04449 (0.016,0.168) We are 95% confident that the procedure we have used for constructing a confidence interval which yield an interval that includes β2. Define the random variables \(X\) and \(P′\), in words. Compare the error bound in part d to the margin of error reported by Gallup. Recently appeared in the mean number of unoccupied seats from a stack of IEEE Spectrum magazines groups, the confidence interval exercises and solutions. The following values: the confidence interval for the population standard deviation is known to called! 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