# confidence interval exercises and solutions

Construct a 90% confidence interval for the population mean time to complete the tax forms. Quick reference. 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. How many male students must you measure? Download this lesson’s related exercise files. Determine the estimated proportion from the sample. 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 population distribution is assumed to be normal. The confidence interval changes but, obviously, the underlying (unknown) mean does not. The mean weight was two ounces with a standard deviation of 0.12 ounces. It is possible that less than half of the population believe this. The answer to “Confidence Intervals. Compare the error bound in part d to the margin of error reported by Gallup. Researchers in a hospital used the drug on a random sample of nine patients. Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. The grams of fat per serving are as follows: 8; 8; 10; 7; 9; 9. The confidence interval can be expressed in terms of a single sample: "There is a 90% probability that the calculated confidence interval from some future experiment encompasses the true value of the population parameter." $$\bar{X}$$ is the mean time to complete tax forms from a sample of 100 customers. Confidence Intervals. Use the formula for $$EBM$$, solved for $$n$$: From the statement of the problem, you know that $$\sigma$$ = 2.5, and you need $$EBM = 1$$. Assume the underlying distribution is approximately normal. Prerequisites. In a population, a random variable follows a normal distribution with an unknown mean and a standard deviation of 2. The confidence level for this study was reported at 95% with a $$\pm 3%$$ margin of error. 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). An article regarding interracial dating and marriage recently appeared in the Washington Post. $$\bar{X}$$ is the mean number of letters sent home from a sample of 20 campers. 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? Suppose that an accounting firm does a study to determine the time needed to complete one person’s tax forms. Determine the confidence interval with a confidence level of 97% for the average population. [ "article:topic", "authorname:openstax", "showtoc:no", "license:ccby" ], 8.7: Confidence Interval -Women's Heights (Worksheet), 8.2: A Single Population Mean using the Normal Distribution, 8.3: A Single Population Mean using the Student t Distribution, 8.6: Confidence Interval (Place of Birth), 8.7: Confidence Interval (Women's Heights). Why? Calculators: Inverse t Distribution: Finds t for a confidence interval. We need to use a Student’s-t distribution, because we do not know the population standard deviation. A sample of Alzheimer's patients are tested to assess the amount of time in stage IV sleep. 11. 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. Confidence Interval Solutions 1. In order to conduct a survey, a sample of 180 workers is selected. 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? percent of all Asians who would welcome a white person into their families. Thus, we do not need as large an interval to capture the true population mean. ), $$n = \frac{z^{2}\sigma^{2}}{EBM^{2}} = \frac{1.812^{2}2.5^{2}}{1^{2}} \approx 20.52$$. The error bound and confidence interval will decrease. Do you think that six packages of fruit snacks yield enough data to give accurate results? $$p′ = \frac{(0.55+0.49)}{2} = 0.52; EBP = 0.55 - 0.52 = 0.03$$. Assume the study was based on n = 94 randomly sampled households throughout Great Britain. Finding Critical Values and Confidence Intervals. What will happen to the level of confidence obtained if 1,000 male Swedes are surveyed instead of 48? No, the confidence interval includes values less than or equal to 0.50. 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. Construct a 95% confidence interval for the population mean worth of coupons. Define the random variables $$X$$ and $$P′$$ in words. (This is the value of $$z$$ for which the area under the density curve to the right of $$z$$ is 0.035. 1.In a sample of 400 selected at random, a sample mean of 50 was obtained. Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car. Confidence Intervals. It was revealed that they used them an average of six months with a sample standard deviation of three months. Use the Student's $$t$$-distribution. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. 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. The mean from the sample is 7.9 with a sample standard deviation of 2.8. 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. For the Halcion versus Dalmane data in Exercise 3, construct a 99% confidence interval estimate of discrepancy in the efficacies of the two drugs. What were the average sales over the nine month period? The following prices were noted: 95, 108, 97, 112, 99, 106, 105, 100, 99, 98, 104, 110, 107, 111, 103, 110. How should she explain the confidence interval to her audience? $$\bar{X}$$ is the mean number of unoccupied seats from a sample of 225 flights. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Forty-eight male Swedes are surveyed. Which distribution should you use for this problem? 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. The following table shows the total receipts during this cycle for a random selection of 20 Leadership PACs. Use a 95% confidence level. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. In Exercises 9–24, construct the confidence interval estimate of the mean. Construct a 98% confidence interval for the population mean weight of the candies. 11-18. 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. percent of all Asians who would welcome a Latino into their families. Explain what this confidence interval means in the context of the problem. You plan to conduct a survey on your college campus to learn about the political awareness of students. The Federal Election Commission (FEC) collects information about campaign contributions and disbursements for candidates and political committees each election cycle. Confidence Interval Exercises. In this survey, 86% of blacks said that they would welcome a white person into their families. 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. e. The confidence interval will decrease in size, because the sample size increased. The sample mean is 23.6 hours. serving size. Therefore, the researchers decide to choose the previous sample using stratified … It is known that 2,500 children, 7,000 adults and 500 elderly live in the neighborhood. The committee randomly surveyed 81 people who recently served as jurors. Suppose that engine displacement is measured in cubic centimeters instead of cubic inches. Forbes magazine published data on the best small firms in 2012. Construct a 90% confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets. The quantity of hemoglobin in the blood spream of a man follows a normal distribution with a standard deviation of 2 g/dl. At SeeTheSolutions.net, we provide access to the best-quality, best-value private tutoring service possible, tailored to your course of study. Explain what a “97% confidence interval” means for this study. The Table shows the ages of the corporate CEOs for a random sample of these firms. $$CL = 0.95 \alpha = 1 - 0.95 = 0.05 z_{\frac{\alpha}{2}} = 1.96$$. Interpret the confidence interval, using complete sentence. All that we would need to do is to take square roots of the endpoints. 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. During the first eight years of the study, 1.5% of the 451 members of the 50-Plus Fitness Association died. 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. Create a 95% confidence interval for the mean total individual contributions. Create a 99% confidence interval for the true proportion of American adults who have illegally downloaded music. Since we are estimating a proportion, given $$P′ = 0.2$$ and $$n = 1000$$, the distribution we should use is $$N\left(0.61, \sqrt{\frac{(0.2)(0.8)}{1000}}\right)$$. If convenient, use technology to construct the confidence interval. What does it mean to be “95% confident” in this problem? Calculate the error bound based on the information provided. The total number of snack pieces in the six bags was 68. Use the Student's t-distribution. If this survey were done by telephone, list three difficulties the companies might have in obtaining random results. Arrow down and enter the name of the list where the data is stored. 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. How many students must you interview? Find the point estimate for the population mean. Author(s) David M. Lane. 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? If we were to sample many groups of nine patients, 95% of the samples would contain the true population mean length of time. Even though the three point estimates are different, do any of the confidence intervals overlap? 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. Number of minutes spent is Stage IV sleep is recorded for sixty-one patients. Construct a 95% confidence interval for the population mean enrollment at community colleges in the United States. To capture the true population mean, we need to have a larger interval. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean. 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. 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)! Solution of exercise Confidence Interval Solutions Solution of exercise 1. Solution: A. Construct a 95% confidence interval for the population proportion who claim they always buckle up. A telephone poll of 1,000 adult Americans was reported in an issue of Time Magazine. 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. In one to three complete sentences, explain what the ±3% represents. The 96% confidence interval is ($47,262,$456,447). Without performing any calculations, describe how the confidence interval would change if the confidence level changed from 99% to 90%. Assume that the population distribution of bag weights is normal. What percentage of (1-alpha)100% confidence intervals for µ will fail to capture µ? The monthly sales of an appliance shop are distributed according to a normal law, with a standard deviation of $900. the effective length of time for a tranquilizer, the mean effective length of time of tranquilizers from a sample of nine patients. The sample size would need to be increased since the critical value increases as the confidence level increases. We wish to construct a 95% confidence interval for the mean height of male Swedes. The sample mean wait time was eight hours with a sample standard deviation of four hours. Unoccupied seats on flights cause airlines to lose revenue. Explain your choice. Exercises 9.2. A camp director is interested in the mean number of letters each child sends during his or her camp session. A sample of 100 bottles yields an average content of 48cl. Stats Lab 8.1 Confidence Interval (Home Costs) Class Time: Names: Student Learning Outcomes The student will calculate the 90% confidence interval fo Consider the regression model developed in Exercise 11-6. Assume the underlying distribution is approximately normal. Construct a 90% confidence interval for the population mean weight of the candies. We are interested in the population proportion of people who feel the president is doing an acceptable job. Missed the LibreFest? The CI for the mean . Assume a population standard deviation of$200. This survey was conducted through automated telephone interviews on May 6 and 7, 2013. It is believed that the standard deviation may have changed from the previous year. 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 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. 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. Can we (with 75% confidence) conclude that at least half of all American adults believe this? to capture µ? Can we (with 95% confidence) conclude that more than half of all American adults believe this? fail. Table shows the total receipts from individuals for a random selection of 40 House candidates rounded to the nearest $100. We are interested in the population proportion of drivers who claim they always buckle up. Find the point estimate and the error bound for this confidence interval. $$n = \frac{z_{\frac{\alpha}{2}}^{2}p'q'}{EPB^{2}} = \frac{1.96^{2}(0.5)(0.5)}{0.05^{2}} = 384.16$$. Large Data Set Exercises. If we took repeated samples, approximately 90% of the samples would produce the same confidence interval. 12. Suppose that the insurance companies did do a survey. Calculate the confidence level for a sample of 12 men which indicates that the population mean blood hemoglobin is between 13 and 15g/dl. $$CL = 0.75$$, so $$\alpha = 1 – 0.75 = 0.25$$ and $$\frac{\alpha}{2} = 0.125 z_{\frac{\alpha}{2}} = 1.150$$. You may want to use the Analysis Lab and various calculators for some of these exercises. Suppose that 14 children, who were learning to ride two-wheel bikes, were surveyed to determine how long they had to use training wheels. $$X =$$ 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. You want to rent an unfurnished one-bedroom apartment in Durham, NC next year. 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”. Assume that the underlying population distribution is normal. 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. According to a recent survey of 1,200 people, 61% feel that the president is doing an acceptable job. 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