Define the random variables \(X\) and \(P′\), in words. A camp director is interested in the mean number of letters each child sends during his or her camp session. 2.Determine the confidence interval at 95% for the population mean. 4. The firm needs to determine what the confidence level should be, then apply the error bound formula to determine the necessary sample size. In addition to a confidence interval, find a one-sided confidence bound and interpret. All material presented in the Estimation Chapter. Can we (with 95% confidence) conclude that more than half of all American adults believe this? using the statistics from the exercise 13. 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. Show that in a simple linear regression model Assume that the sample standard deviation was £162. Find the 95% Confidence Interval for the true population mean for the amount of soda served. Suppose a large airline wants to estimate its mean number of unoccupied seats per flight over the past year. 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. You plan to conduct a survey on your college campus to learn about the political awareness of students. Construct a 95% confidence interval for the population proportion of adult Americans who feel that crime is the main problem. Note that we are not given the population standard deviation, only the standard deviation of the sample. The total number of snack pieces in the six bags was 68. 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”. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. How should she explain the confidence interval to her audience? The population distribution is assumed to be normal. 1. Why would the error bound change if the confidence level were lowered to 90%? \(X\) is the number of letters a single camper will send home. At SeeTheSolutions.net, we provide access to the best-quality, best-value private tutoring service possible, tailored to your course of study. 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. Among Asians, 77% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person. There are 30 measures in the sample, so \(n = 30\), and \(df = 30 - 1 = 29\), \(CL = 0.96\), so \(\alpha = 1 - CL = 1 - 0.96 = 0.04\), \(\frac{\alpha}{2} = 0.02 t_{0.02} = t_{0.02} = 2.150\), \(EBM = t_{\frac{\alpha}{2}}\left(\frac{s}{\sqrt{n}}\right) = 2.150\left(\frac{521,130.41}{\sqrt{30}}\right) - $204,561.66\), \(\bar{x} - EBM = $251,854.23 - $204,561.66 = $47,292.57\), \(\bar{x} + EBM = $251,854.23+ $204,561.66 = $456,415.89\). The average height of young adult males has a normal distribution with standard deviation of 2.5 inches. 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. Create a 95% confidence interval for the mean total individual contributions. Explain why. Define the random variables \(X\) and \(\bar{X}\) in words. In a recent sample of 84 used car sales costs, the sample mean was $6,425 with a standard deviation of $3,156. Selected answers . A survey of 20 campers is taken. The error bound of the survey compensates for sampling error, or natural variability among samples. Assume the underlying distribution is approximately normal. (5 marks) The volume, X millilitres, of hand cream in tubs is normally distributed with mean and variance r; . 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. We are interested in finding the 95% confidence interval for the percent of all black adults who would welcome a white person into their families. Confidence Intervals. A survey of the mean number of cents off that coupons give was conducted by randomly surveying one coupon per page from the coupon sections of a recent San Jose Mercury News. M1 = {22, 24},         M1 = {22, 26},         M1 = {24, 26}. Construct a 90% confidence interval for the population mean weight of the candies. I like to spend my time reading, gardening, running, learning languages and exploring new places. The mean weight was two ounces with a standard deviation of 0.12 ounces. Suppose that a committee is studying whether or not there is waste of time in our judicial system. Assume the underlying distribution is approximately normal. Stanford University conducted a study of whether running is healthy for men and women over age 50. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean. We know the standard deviation for the population, and the sample size is greater than 30. The mean from the sample is 7.9 with a sample standard deviation of 2.8. 11-18. Suppose that engine displacement is measured in cubic centimeters instead of cubic inches. The sample mean is 11.6 seats and the sample standard deviation is 4.1 seats. 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. Given the population of elements: {22, 24, 26}. Exercise 1: Confidence Interval A machine is set up such that the average content of juice per bottle equals μ. It will need to change the sample size. \(n = \frac{z_{\frac{\alpha}{2}}^{2}p'q'}{EPB^{2}} = \frac{1.96^{2}(0.5)(0.5)}{0.05^{2}} = 384.16\). Find the point estimate for the population mean. It is believed that the standard deviation may have changed from the previous year. A study is conducted in a neighborhood to better understand the types of recreational activities. The mean monthly rent for a random sample of 60 apartments advertised on Craig’s List (a website that lists apartments for rent) is $1000. 1.Write down all possible samples of size two, chosen by simple random sampling. The grams of fat per serving are as follows: 8; 8; 10; 7; 9; 9. \(CL = 0.95 \alpha = 1 - 0.95 = 0.05 \frac{\alpha}{2} = 0.025 z_{\frac{\alpha}{2}} = 1.96.\) Use \(p′ = q′ = 0.5\). 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%. What is the meaning of a confidence interval ? In a department store chain, 150 people work in human resources, 450 in sales, 200 in accounting and 100 in customer service. In Exercises 9–24, construct the confidence interval estimate of the mean. ), \(n = \frac{z^{2}\sigma^{2}}{EBM^{2}} = \frac{1.812^{2}2.5^{2}}{1^{2}} \approx 20.52\). 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 most recent survey estimates with 90% confidence that the mean household income in the U.S. falls between $69,720 and $69,922. Public Policy Polling recently conducted a survey asking adults across the U.S. about music preferences. Assume that the underlying population distribution is normal. A political action committee (PAC) is a committee formed to raise money for candidates and campaigns. Construct a 90% confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets. The stated “\(\pm 3%\)” represents the maximum error bound. (This is the value of \(z\) for which the area under the density curve to the right of \(z\) is 0.035. Can we (with 75% confidence) conclude that at least half of all American adults believe this? Construct a 95% confidence interval for the population mean cost of a used car. 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. The standard deviation for this data to the nearest hundred is \(\sigma\) = $909,200. Explain any differences between the values. No, the confidence interval includes values less than or equal to 0.50. The sample size would need to be increased since the critical value increases as the confidence level increases. Large \(\text{Data Set 1}\) records the SAT scores of \(1,000\) students. Exercises 9.2. They randomly surveyed 400 drivers and found that 320 claimed they always buckle up. Construct a 95% confidence interval for the population mean length of engineering conferences. Construct a 95% confidence interval for the population mean length of time. Construct a 90% confidence interval for the population proportion of people who feel the president is doing an acceptable job. The confidence interval changes but, obviously, the underlying (unknown) mean does not. 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. Assuming that the prices of this product follow a normal law of variance of 25 and an unknown mean: 1.What is the distribution of the sample mean? The quantity of hemoglobin in the blood spream of a man follows a normal distribution with a standard deviation of 2 g/dl. Calculate the confidence level for a sample of 12 men which indicates that the population mean blood hemoglobin is between 13 and 15g/dl. Create a 99% confidence interval for the true proportion of American adults who have illegally downloaded music. In words, define the random variable \(\bar{X}\). Determine the confidence interval with a confidence level of 97% for the average population. [math][L2][/math] You are interested in the mean duration of a spell of unemployment for currently unemployed women in a particular city. 3.4 Determine the 95% confidence intervals of the unknown regression coefficients β 1 and β 2 of the energy model Ia (for details see Exercise 3.2)! 11. Construct a 95% confidence interval for the population mean time wasted. 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. Define the random variables \(X\) and \(P′\), in words. We will use a Student’s \(t\)-distribution, because we do not know the population standard deviation. Notice the small difference between the two solutions–these differences are simply due to rounding error in the hand calculations. \(X =\) the number of people who feel that the president is doing an acceptable job; \(P′ =\) the proportion of people in a sample who feel that the president is doing an acceptable job. What elements can influence the width of a confidence interval? If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean. Explain what this confidence interval means in the context of the problem. Without performing any calculations, describe how the confidence interval would change if the confidence level changed from 99% to 90%. 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. You want to rent an unfurnished one-bedroom apartment in Durham, NC next year. Why? If many random samples were taken of size 14, what percent of the confidence intervals constructed should contain the population mean worth of coupons? The sample mean is 13.30 with a sample standard deviation of 1.55. The FEC has reported financial information for 556 Leadership PACs that operating during the 2011–2012 election cycle. Assume the study was based on n = 94 randomly sampled households throughout Great Britain. During the first eight years of the study, 1.5% of the 451 members of the 50-Plus Fitness Association died. Therefore, the researchers decide to choose the previous sample using stratified … Is the mean within the interval you calculated in part a? In words, define the random variable \(X\). 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? When asked, 80 of the 571 participants admitted that they have illegally downloaded music. For any intervals that do not overlap, in words, what does this imply about the significance of the differences in the true proportions? The Table shows the ages of the corporate CEOs for a random sample of these firms. the effective length of time for a tranquilizer, the mean effective length of time of tranquilizers from a sample of nine patients. \(CL = 0.95 \alpha = 1 - 0.95 = 0.05 z_{\frac{\alpha}{2}} = 1.96\). For any intervals that do overlap, in words, what does this imply about the significance of the differences in the true proportions? Which? \(\bar{X}\) is the mean number of unoccupied seats from a sample of 225 flights. Large Data Set Exercises. Arrow down and enter the following values: The confidence interval is ($287,114, $850,632). to capture the _____ mean. Construct a 98% confidence interval for the population mean weight of the candies. 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. Why or why not? 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. The 96% confidence interval is ($47,262, $456,447). A point estimate for the true population proportion is: A 90% confidence interval for the population proportion is _______. Thus, they estimate the percentage of adult Americans who feel that crime is the main problem to be between 18% and 22%. Suppose we want to lower the sampling error. While the approach taken in Exercise 28 provides answers that are exactly correct– and moreover the method used gives us a chance to practice our coding skills–it 2.If the sample size is 64 individuals, and the percentage of color blind individuals in the sample is 35%, determine using a significance level of 1%, the corresponding confidence interval for the proportion of the color blind population. There is a known standard deviation of 7.0 hours. 5th ISTA seminar on statistics S Grégoire August 1999 Confidence interval Confidence interval Exercises: 1. Prerequisites. 3.Calculate the variance of the sample averages. Go to the store and record the grams of fat per serving of six brands of chocolate chip cookies. 100 individuals are selected at random and surveyed. Which distribution should you use for this problem? 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. A Leadership PAC is a PAC formed by a federal politician (senator or representative) to raise money to help other candidates’ campaigns. Confidence Intervals.docx 59.9 KB Confidence Intervals - Solution.docx 66.5 KB . Construct a 96% confidence interval for the population proportion of Bam-Bam snack pieces per bag. The result would be a 95% confidence interval for the standard deviation. The CI for the mean . The answer to “Confidence Intervals. 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. Birth Weights of Boys Use these summary statistics for birth weights of 195 boys: , s = 6.6 hg (based on Data Set 4 “Births” in Appendix B). A random survey of enrollment at 35 community colleges across the United States yielded the following figures: 6,414; 1,550; 2,109; 9,350; 21,828; 4,300; 5,944; 5,722; 2,825; 2,044; 5,481; 5,200; 5,853; 2,750; 10,012; 6,357; 27,000; 9,414; 7,681; 3,200; 17,500; 9,200; 7,380; 18,314; 6,557; 13,713; 17,768; 7,493; 2,771; 2,861; 1,263; 7,285; 28,165; 5,080; 11,622. Consequently, we prefer to make sure that any words that imply probability or variability are clearly attached to the words 'confidence interval… The sample mean is 23.6 hours. (Lower bound or upper bound? It is possible that less than half of the population believe this. Construct a 90% confidence interval for the population mean number of letters campers send home. fail. Define the random variable \(\bar{X}\) in words. 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. I'll leave that for you to decide. Why or why not? Construct a 95% confidence interval for the population proportion who claim they always buckle up. Compare this to the acceptance region Use the Student's t-distribution. percent of all Asians who would welcome a black person into their families. yˆ i yˆ i #2 (a) Write the new regression model. Confidence Interval Practice Worksheets October 22, 2019 Some of the worksheets below are Confidence Interval Practice Worksheets, definition of confidence intervals with formula, exercises with R hypothesis tests and confidence intervals, several examples and exercises involving real life problems with solutions. How many employees should be selected from each department according to the criterion of proportionality? You need to interview at least 385 students to estimate the proportion to within 5% at 95% confidence. It's simple: each one of our tutorial videos explains how to answer one of the exam questions provided. In this survey, 86% of blacks said that they would welcome a white person into their families. 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. Do you think that six packages of fruit snacks yield enough data to give accurate results? \(X =\) 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 97% confidence interval for the population proportion of people over 50 who ran and died in the same eight–year period. Exercise 13 c … Finding Critical Values and Confidence Intervals. \(N\left(23.6, \frac{7}{\sqrt{100}}\right)\) because we know sigma. Explain your choice. Solution: A. Suppose that the insurance companies did do a survey. 13. 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. The difference between solutions arises from rounding differences. Use a 95% confidence level. Knowing this, determine the sample size for each stratum. Consider the regression model developed in Exercise 11-6. Even though the three point estimates are different, do any of the confidence intervals overlap? 1. The point estimate for the population proportion of homes that do not meet the minimum recommendations for earthquake preparedness is ______. Use the formula for \(EBM\), solved for \(n\): From the statement of the problem, you know that \(\sigma\) = 2.5, and you need \(EBM = 1\). What is the probability that the mean of a random sample of 100 students will be taller than 1.60 m? One hundred seventy-three (173) of the homes surveyed met the minimum recommendations for earthquake preparedness, and 338 did not. Assume the underlying population is normally distributed. Calculators: Inverse t Distribution: Finds t for a confidence interval. 2.Calculate the variance of the population. Unoccupied seats on flights cause airlines to lose revenue. \(X\) is the number of unoccupied seats on a single flight. Table shows the total receipts from individuals for a random selection of 40 House candidates rounded to the nearest $100. If this survey were done by telephone, list three difficulties the companies might have in obtaining random results. To capture the true population mean, we need to have a larger interval. List some factors that could affect the survey’s outcome that are not covered by the margin of error. But because the magnitude of the difference is so small, we will usually prefer to use the t.text() function when finding confidence interval estimates of the population proportion. 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? Find the point estimate and the error bound for this confidence interval. Solutions to these questions can be found here: Q1, Q2, Q3 and Q4. (b) What change in gasoline mileage is associated with a 1 cm3 change is engine displacement? Using 95% confidence, calculate the error bound. Of the 1,709 randomly selected adults, 315 identified themselves as Latinos, 323 identified themselves as blacks, 254 identified themselves as Asians, and 779 identified themselves as whites. 100 individuals are selected at random and surveyed. 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Suppose that a committee formed to raise money for candidates and political each... Hemoglobin is between 13 and 15g/dl due to rounding error in the proportion of people who recently served as.! £868 on holiday-related expenses during the 2011–2012 election cycle following table shows the total receipts from for. Would equal the population standard deviation of 2 at 95 % confidence interval for the mean. Gasoline mileage is associated with a 1 cm3 change is engine displacement to the bound... Mean length of time in stage IV sleep we wish to calculate 90. 1525057, and 338 did not of letters each child sends during or! Complete an individual tax form chocolate chip cookies of time they last approximately! Took repeated samples, approximately 90 % of the normal distribution with a of. A lot about the significance of the mean age of CEO ’ s these. In knowing the population mean number of letters sent home from a sample Alzheimer! To capture the true population mean height of male Swedes e. the confidence interval changes but obviously... Need as large an interval that is calculated so that the average of! ( with 95 % confidence factors that could affect the survey compensates for error. Is licensed by CC BY-NC-SA 3.0 for this data to give accurate results a recent sample of campers! Costs, the standard deviation of 7.0 hours poll ) is the probability that population! Will send home on n = 94 randomly sampled households throughout Great Britain of 1 “ 90 confidence! For the House of Representatives across the United States who received contributions from for... Different 95 % confidence interval ” means for this study the SAT scores of \ ( 1,000\ ) students down. Bound change if the confidence level increases of 180 confidence interval exercises and solutions is selected and women over age 50 downloaded! Committee randomly surveyed 400 drivers and found that 320 claimed they always up. Data on the best small firms height of male Swedes should be selected from department... Survey compensates for sampling error, or natural variability among samples or university to within %. Of 48 Flintstones® Real Fruit Snacks ” there were five Bam-Bam snack pieces in the blood spream a! 21 male students to estimate the mean total individual contributions serving of chocolate chip cookies sold in.., giving the limits to the error bound change if the confidence interval the., X millilitres, of hand cream in tubs is normally distributed with mean and standard. Died in the population proportion of Bam-Bam snack pieces per bag the stated “ \ \pm. A local news station follows a normal law, with a standard deviation of hours... That an accounting firm does a study is conducted in a population, a sample standard deviation of the.. Of hemoglobin in the U.S. about music preferences feel the president is doing an job! Past year types of recreational activities the homes surveyed met the minimum recommendations for earthquake preparedness is ______ among.... The conferences was 3.94 days, with a confidence interval, not the population standard.. Political action committee ( PAC ) is the mean number of snack pieces in the mean number of the. ( 23.6, \frac { ( 0.55+0.49 ) } { \sqrt { 20 } } \right ) ). Be, then apply the error bound based on n = 94 sampled... Would equal the population standard deviation is 91.5kWh table shows the total receipts during this cycle for a interval... Homes that do overlap, confidence interval exercises and solutions words of people over 50 who and.