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. The sample mean is 11.6 seats and the sample standard deviation is 4.1 seats. Since we increase the confidence level, we need to increase either our error bound or the sample size. Find the point estimate for the population mean. The stated “$$\pm 3%$$” represents the maximum error bound. Thus, we do not need as large an interval to capture the true population mean. In one to three complete sentences, explain what the ±3% represents. It was revealed that they used them an average of six months with a sample standard deviation of three months. Suppose that a committee is studying whether or not there is waste of time in our judicial system. $$X =$$ 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. percent of all Asians who would welcome a white person into their families. Salwa Al Mauly 2017002901 Solutions of Confidence Interval Exercises 1. sd/ sqrt (n) ( 322/ sqrt ( 100) ) = 32.2 1.984 * 32.2 = 63.885 [ 8500 ± 63.885 ] = [ 8436.115 , 8563.885 ] 2. Construct a 99% confidence interval for the population mean length of time using training wheels. 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. Construct a 95% confidence Interval for 19, giving the limits to the nearest integer. Explain your choice. Since we are estimating a proportion, given $$P′ = 0.2$$ and $$n = 1000$$, the distribution we should use is $$N\left(0.61, \sqrt{\frac{(0.2)(0.8)}{1000}}\right)$$. The mean weight was two ounces with a standard deviation of 0.12 ounces. Suppose that engine displacement is measured in cubic centimeters instead of cubic inches. 13. 12. It is assumed that the distribution for the length of time they last is approximately normal. Confidence Intervals. 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 Finding Critical Values and Confidence Intervals. 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. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean. (b) What change in gasoline mileage is associated with a 1 cm3 change is engine displacement? Construct a 95% confidence interval for the population proportion of adult Americans who feel that crime is the main problem. 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. 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. Therefore, the researchers decide to choose the previous sample using stratified … The population distribution is assumed to be normal. The standard deviation for this data to the nearest hundred is $$\sigma$$ =$909,200. What is the meaning of a confidence interval ? For any intervals that do not overlap, in words, what does this imply about the significance of the differences in the true proportions? One hundred seventy-three (173) of the homes surveyed met the minimum recommendations for earthquake preparedness, and 338 did not. 100 individuals are selected at random and surveyed. 2. Suppose that an accounting firm does a study to determine the time needed to complete one person’s tax forms. There are 30 measures in the sample, so $$n = 30$$, and $$df = 30 - 1 = 29$$, $$CL = 0.96$$, so $$\alpha = 1 - CL = 1 - 0.96 = 0.04$$, $$\frac{\alpha}{2} = 0.02 t_{0.02} = t_{0.02} = 2.150$$, $$EBM = t_{\frac{\alpha}{2}}\left(\frac{s}{\sqrt{n}}\right) = 2.150\left(\frac{521,130.41}{\sqrt{30}}\right) - 204,561.66$$, $$\bar{x} - EBM = 251,854.23 - 204,561.66 = 47,292.57$$, $$\bar{x} + EBM = 251,854.23+ 204,561.66 = 456,415.89$$. $$X =$$ 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. Calculate the confidence level for a sample of 12 men which indicates that the population mean blood hemoglobin is between 13 and 15g/dl. Exercise 1: Confidence Interval A machine is set up such that the average content of juice per bottle equals µ. Stanford University conducted a study of whether running is healthy for men and women over age 50. Large Data Set Exercises. 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. serving size. Did you expect it to be? How many students must you interview? Exercise 1: Confidence Interval A machine is set up such that the average content of juice per bottle equals μ. 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. 2.Calculate the variance of the population. percent of all Asians who would welcome a Latino into their families. 1.In a sample of 400 selected at random, a sample mean of 50 was obtained. 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%. 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. The confidence interval changes but, obviously, the underlying (unknown) mean does not. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. 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. t Distribution: Computes areas of the t distribution. 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). Can we (with 95% confidence) conclude that more than half of all American adults believe this? I am passionate about travelling and currently live and work in Paris. Construct a 95% confidence interval for the population mean length of engineering conferences. Compare this to the acceptance region What percentage of 95% confidence intervals for µ will . Create a 95% confidence interval for the mean total individual contributions. We are interested in the population proportion of drivers who claim they always buckle up. Solution: A. We know the standard deviation for the population, and the sample size is greater than 30. Construct a 96% confidence interval for the population proportion of Bam-Bam snack pieces per bag. e. The confidence interval will decrease in size, because the sample size increased. What is one way to accomplish that? The average height of young adult males has a normal distribution with standard deviation of 2.5 inches. For any intervals that do overlap, in words, what does this imply about the significance of the differences in the true proportions? 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. For the Halcion versus Dalmane data in Exercise 3, construct a 99% confidence interval estimate of discrepancy in the efficacies of the two drugs. Define the random variable $$X$$ in words. Among Asians, 77% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person. Use this sample data to construct a 90% confidence interval for the mean age of CEO’s for these top small firms. Quick reference. 1.Write down all possible samples of size two, chosen by simple random sampling. We will use a Student’s $$t$$-distribution, because we do not know the population standard deviation. 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. During the 2012 campaign season, there were 1,619 candidates for the House of Representatives across the United States who received contributions from individuals. Which distribution should you use for this problem? This survey was conducted through automated telephone interviews on May 6 and 7, 2013. In a recent sample of 84 used car sales costs, the sample mean was $6,425 with a standard deviation of$3,156. There is a known standard deviation of 7.0 hours. These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax. Create a confidence interval for the results of this study. $$X$$ is the number of unoccupied seats on a single flight. $$N 7.9\left(\frac{2.5}{\sqrt{20}}\right)$$. Confidence Interval and Hypothesis Testing: Exercises and Solutions You can use the graphical representation of the normal distribution to solve the problems. $$p′ = \frac{(0.55+0.49)}{2} = 0.52; EBP = 0.55 - 0.52 = 0.03$$. The following table shows the total receipts during this cycle for a random selection of 20 Leadership PACs. Large Data Set missing from the original. Thus, they estimate the percentage of adult Americans who feel that crime is the main problem to be between 18% and 22%. Even though the three point estimates are different, do any of the confidence intervals overlap? What is meant by the term “90% confident” when constructing a confidence interval for a mean? Consequently, we prefer to make sure that any words that imply probability or variability are clearly attached to the words 'confidence interval… Stats Lab 8.1 Confidence Interval (Home Costs) Class Time: Names: Student Learning Outcomes The student will calculate the 90% confidence interval fo The error bound of the survey compensates for sampling error, or natural variability among samples. The 96% confidence interval is ($47,262,$456,447). Why or why not? 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? The sample size would need to be increased since the critical value increases as the confidence level increases. $$N\left(23.6, \frac{7}{\sqrt{100}}\right)$$ because we know sigma. $$CL = 0.95 \alpha = 1 - 0.95 = 0.05 \frac{\alpha}{2} = 0.025 z_{\frac{\alpha}{2}} = 1.96.$$ Use $$p′ = q′ = 0.5$$. If the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? It concluded with 95% confidence that 49% to 55% of Americans believe that big-time college sports programs corrupt the process of higher education. It is known that 2,500 children, 7,000 adults and 500 elderly live in the neighborhood. 11-18. The answer to “Confidence Intervals. Assume the underlying population is normal. Knowing this, determine the sample size for each stratum. Exercise 13 c … What will happen to the level of confidence obtained if 1,000 male Swedes are surveyed instead of 48? 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 population standard deviation is known to be 0.1 ounce. Construct a 92% confidence interval for the population mean number of unoccupied seats per flight. What is the probability that the mean of a random sample of 100 students will be taller than 1.60 m? to capture µ? If convenient, use technology to construct the confidence interval. Can we (with 75% confidence) conclude that at least half of all American adults believe this? $$n = \frac{z_{\frac{\alpha}{2}}^{2}p'q'}{EPB^{2}} = \frac{1.96^{2}(0.5)(0.5)}{0.05^{2}} = 384.16$$. Normal. It is known that 2,500 children, 7,000 adults and 500 elderly live in the neighborhood. Confidence Interval Solutions 1. Determine the estimated proportion from the sample. Calculators: Inverse t Distribution: Finds t for a confidence interval. 3. Suppose that 14 children, who were learning to ride two-wheel bikes, were surveyed to determine how long they had to use training wheels. Define the random variables $$X$$ and $$P′$$ in words. Why? Compare the error bound in part d to the margin of error reported by Gallup. How can I visualise by a drawing a confidence interval? Forbes magazine published data on the best small firms in 2012. In order to conduct a survey, a sample of 180 workers is selected. You need to interview at least 385 students to estimate the proportion to within 5% at 95% confidence. Number of minutes spent is Stage IV sleep is recorded for sixty-one patients. (Lower bound or upper bound? In six packages of “The Flintstones® Real Fruit Snacks” there were five Bam-Bam snack pieces. Do you think that six packages of fruit snacks yield enough data to give accurate results? Define the random variables $$X$$ and $$P′$$, in words. 1. All that we would need to do is to take square roots of the endpoints. A study is conducted in a neighborhood to better understand the types of recreational activities. Construct a 95% confidence interval. Exercises 9.2. The most recent survey estimates with 90% confidence that the mean household income in the U.S. falls between $69,720 and$69,922. Construct a 95% confidence interval for the population mean worth of coupons. 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. Explain what this confidence interval means in the context of the problem. 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. The sample mean is 13.30 with a sample standard deviation of 1.55. slightly different 95% confidence interval to the first experiment. According to the error bound formula, the firm needs to survey 206 people. Assume the underlying population is normal. 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. 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. One way to lower the sampling error is to increase the sample size. Define the random variables $$X$$ and $$\bar{X}$$ in words. When inferring statistical values based upon a sample, there is a band of uncertainty around the sample statistic in which the population statistic lies. It is interested in the mean amount of time individuals waste at the courthouse waiting to be called for jury duty. 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? It is believed that the standard deviation may have changed from the previous year. We need to use a Student’s-t distribution, because we do not know the population standard deviation. 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. 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. $$CL = 0.95 \alpha = 1 - 0.95 = 0.05 z_{\frac{\alpha}{2}} = 1.96$$. Explain what a “97% confidence interval” means for this study. Interpret the confidence interval, using complete sentence. Construct a 90% confidence interval for the population mean number of letters campers send home. Watch the recordings here on Youtube! 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. A Leadership PAC is a PAC formed by a federal politician (senator or representative) to raise money to help other candidates’ campaigns. 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)! Forty-eight male Swedes are surveyed. Suppose a large airline wants to estimate its mean number of unoccupied seats per flight over the past year. The sample standard deviation is 2.8 inches. b. Repeat part (a) for a 99% confidence... View Answer. We are interested in the proportion of people over 50 who ran and died in the same eight-year period. A reporter is covering the release of this study for a local news station. It is possible that less than half of the population believe this. A sample of 16 small bags of the same brand of candies was selected. Explain any differences between the values. In a population, a random variable follows a normal distribution with an unknown mean and a standard deviation of 2. (f) An interval that is calculated so that it has known likelihood of capturing the parameter. A political action committee (PAC) is a committee formed to raise money for candidates and campaigns. 4. A sample of 100 bottles yields an average content of 48cl. Confidence Intervals. The result would be a 95% confidence interval for the standard deviation. 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? 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. The Table shows the ages of the corporate CEOs for a random sample of these firms. If we took repeated samples, approximately 90% of the samples would produce the same confidence interval. $$\bar{X}$$ is the mean time to complete tax forms from a sample of 100 customers. 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”. 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. Construct three 95% confidence intervals. According to a recent survey of 1,200 people, 61% feel that the president is doing an acceptable job. $$\bar{X}$$ is the mean number of letters sent home from a sample of 20 campers. Use the Student's $$t$$-distribution. The grams of fat per serving are as follows: 8; 8; 10; 7; 9; 9. Using 95% confidence, calculate the error bound. Assume a population standard deviation of \$200. Consider the regression model developed in Exercise 11-6. Exercises and Solutions You can use the graphical representation of the normal distribution to solve the problems. 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. Confidence Intervals. If we were to sample many groups of nine patients, 95% of the samples would contain the true population mean length of time. You plan to conduct a survey on your college campus to learn about the political awareness of students. We are interested in the population proportion of people who feel the president is doing an acceptable job. 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. The point estimate for the population proportion of homes that do not meet the minimum recommendations for earthquake preparedness is ______. Explain in a complete sentence what the confidence interval means. If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples. Selected answers . Find the point estimate and the error bound for this confidence interval. Arrow down and enter the name of the list where the data is stored. Assume that the sample standard deviation was £162. How would the number of people the firm surveys change? It is known that 2,500 children, 7,000 adults and 500 elderly live in the, Solution of exercise Confidence Interval Solutions. 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