construct a 90% confidence interval for the population mean
The first solution is shown step-by-step (Solution A). \(n = \dfrac{z^{2}\sigma^{2}}{EBM^{2}} = \dfrac{(1.96)^{2}(15)^{2}}{2^{2}}\) using the sample size equation. STAT TESTS A: 1-PropZinterval with \(x = (0.52)(1,000), n = 1,000, CL = 0.75\). The second solution uses the TI-83, 83+, and 84+ calculators (Solution B). Kuczmarski, Robert J., Cynthia L. Ogden, Shumei S. Guo, Laurence M. Grummer-Strawn, Katherine M. Flegal, Zuguo Mei, Rong Wei, Lester R. Curtin, Alex F. Roche, Clifford L. Johnson. Every cell phone emits RF energy. Arrow down and enter the following values: The confidence interval is ($287,114, $850,632). This means that we can proceed with finding a 95% confidence interval for the population variance. If we know the confidence interval, we can work backwards to find both the error bound and the sample mean. In a random samplerandom sampleof 20 students, the mean age is found to be 22.9 years. \(\bar{X}\) is normally distributed, that is, \(\bar{X} \sim N(\mu_{x},\dfrac{\sigma}{\sqrt{n}})\). Refer to Exercise. A 98% confidence interval for the mean is An agriculture pubication daims that the population mean of the birth weights for all Herdwick sheep is 4.54 kg. Arrow down to 7:ZInterval. These intervals are different for several reasons: they were calculated from different samples, the samples were different sizes, and the intervals were calculated for different levels of confidence. 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. However, sometimes when we read statistical studies, the study may state the confidence interval only. Use the Student's t-distribution. A. 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 \(\sigma\). State the confidence interval. \[CL + \dfrac{\alpha}{2} + \dfrac{\alpha}{2} = CL + \alpha = 1.\nonumber \], The interpretation should clearly state the confidence level (\(CL\)), explain what population parameter is being estimated (here, a population mean), and state the confidence interval (both endpoints). Available online at www.fec.gov/finance/disclosuresummary.shtml (accessed July 2, 2013). Construct a 90 % confidence interval to estimate the population mean using the accompanying data. We need to find the value of \(z\) that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution \(Z \sim N(0, 1)\). You plan to conduct a survey on your college campus to learn about the political awareness of students. \(CL = 0.95 \alpha = 1 - 0.95 = 0.05 \frac{\alpha}{2} = 0.025 z_{\frac{\alpha}{2}} = 1.96.\) Use \(p = q = 0.5\). Find the point estimate for mean U.S. household income and the error bound for mean U.S. household income. Why would the error bound change if the confidence level were lowered to 90%? Thus, we do not need as large an interval to capture the true population mean. \[\dfrac{\alpha}{2} = \dfrac{1 - CL}{2} = \dfrac{1 - 0.93}{2} = 0.035\nonumber \], \[EBM = (z_{0.035})\left(\dfrac{\sigma}{\sqrt{n}}\right) = (1.812)\left(\dfrac{0.337}{\sqrt{20}}\right) = 0.1365\nonumber \], \[\bar{x} - EBM = 0.940 - 0.1365 = 0.8035\nonumber \], \[\bar{x} + EBM = 0.940 + 0.1365 = 1.0765\nonumber \]. Use a sample size of 20. Construct a 90% confidence interval for the population mean grade point average. Please enter the necessary parameter values, and then click 'Calculate'. Find a 90% confidence interval estimate for the population mean delivery time. Assume the underlying distribution is approximately normal. If we decrease the sample size \(n\) to 25, we increase the error bound. 9.1 - Confidence Intervals for a Population Proportion A random sample is gathered to estimate the percentage of American adults who believe that parents should be required to vaccinate their children for diseases like measles, mumps, and rubella. The population standard deviation for the height of high school basketball players is three inches. In Equation \ref{samplesize}, \(z\) is \(z_{\dfrac{a}{2}}\), corresponding to the desired confidence level. Find the point estimate and the error bound for this confidence interval. \(\bar{x} - EBM = 1.024 0.1431 = 0.8809\), \(\bar{x} - EBM = 1.024 0.1431 = 1.1671\). \[z_{\dfrac{\alpha}{2}} = z_{0.05} = 1.645\nonumber \]. We know the sample mean but we do not know the mean for the entire population. Your email address will not be published. A sample of 16 small bags of the same brand of candies was selected. Construct a 95% confidence interval for the population mean worth of coupons. Suppose we know that a confidence interval is (42.12, 47.88). Unoccupied seats on flights cause airlines to lose revenue. X = 46 o = 12 n42 With 99% confidence, when n = 42 the population mean is between a lower limit of (Round to two decimal places as needed.) and an upper limit of Construct a 95% confidence interval to estimate the population mean with X = 102 and o = 25 . The reason that we would even want to create a confidence interval for a mean is because we want to capture our uncertainty when estimating a population mean. What is the error bound? Arrow down to Calculate and press ENTER. If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size. 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. Assuming a population standard deviation of 0.2 mph, construct a 90% confidence interval for the mean difference between true speed and indicated speed for all vehicles. One hundred seventy-three (173) of the homes surveyed met the minimum recommendations for earthquake preparedness, and 338 did not. Assume the underlying distribution is approximately normal. The committee randomly surveyed 81 people who recently served as jurors. With a 90 percent confidence interval, you have a 10 percent chance of being wrong. Available online at. Of the 1,027 U.S. adults randomly selected for participation in the poll, 69% thought that it should be illegal. Suppose that an accounting firm does a study to determine the time needed to complete one persons tax forms. Explain any differences between the values. Why? Here, the margin of error (\(EBM\)) is called the error bound for a population mean (abbreviated EBM). The confidence interval estimate will have the form: \[(\text{point estimate} - \text{error bound}, \text{point estimate} + \text{error bound})\nonumber \], \[(\bar{x} - EBM, \bar{x} + EBM)\nonumber \]. We estimate with 90% confidence that the true population mean exam score for all statistics students is between 67.18 and 68.82. What will happen to the error bound obtained if 1,000 male Swedes are surveyed instead of 48? If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean. \(\sigma = 3; n = 36\); The confidence level is 95% (CL = 0.95). 7,10,10,4,4,1 Complete parts a and b. a. Construct a 90% confidence interval for the population mean . The effects of these kinds of changes are the subject of the next section in this chapter. percent of all Asians who would welcome a white person into their families. Assume that the population distribution of bag weights is normal. Create a confidence interval for the results of this study. The 90% confidence interval is (67.18, 68.82). 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. To capture the central 90%, we must go out 1.645 "standard deviations" on either side of the calculated sample mean. Arrow down and enter the following values: The confidence interval is (to three decimal places) (0.881, 1.167). The difference between solutions arises from rounding differences. This is the t*- value for a 95 percent confidence interval for the mean with a sample size of 10. The area to the right of \(z_{0.05}\) is \(0.05\) and the area to the left of \(z_{0.05}\) is \(1 - 0.05 = 0.95\). 3. It is denoted by. C. Construct a 95% confidence interval for the population mean enrollment at community colleges in the United States. For a two-tailed 95% confidence interval, the alpha value is 0.025, and the corresponding critical value is 1.96. The population standard deviation is known to be 2.5. The area to the right of \(z_{0.025}\) is 0.025 and the area to the left of \(z_{0.025}\) is \(1 0.025 = 0.975\). The mean weight was two ounces with a standard deviation of 0.12 ounces. The value 1.645 is the z-score from a standard normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail. Use this sample data to construct a 90% confidence interval for the mean age of CEO's for these top small firms. Remember, in this section we already know the population standard deviation \(\sigma\). It is denoted by n. Define the random variables \(X\) and \(\bar{X}\) in words. Explain what a 95% confidence interval means for this study. The firm needs to determine what the confidence level should be, then apply the error bound formula to determine the necessary sample size. The sample mean is 23.6 hours. Each of the tails contains an area equal to \(\dfrac{\alpha}{2}\). If we don't know the error bound: \(\bar{x} = \dfrac{(67.18+68.82)}{2} = 68\). Researchers in a hospital used the drug on a random sample of nine patients. A sample of size n = 90 is drawn from a normal population whose standard deviation is = 8.5.The sample mean is x = 36.76.Part: 0/2 Part 1 of 2 (a) Construct a 98% confidence interval for .Round the answer to at least two decimal places. Ninety-five percent of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score. Interpret the confidence interval in the context of the problem. Construct a 90% confidence interval for the mean GPA of all students at the university. Suppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of six minutes. American Fact Finder. U.S. Census Bureau. 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. Arrow down and enter three for , 68 for \(\bar{x}\), 36 for \(n\), and .90 for C-level. The reporter claimed that the poll's " margin of error " was 3%. Of course, other levels of confidence are possible. Now plug in the numbers: This page titled 7.2: Confidence Intervals for the Mean with Known Standard Deviation 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. It means that should you repeat an experiment or survey over and over again, 95 percent of the time your results will match the results you get from a population (in other words, your statistics would be sound! During the first eight years of the study, 1.5% of the 451 members of the 50-Plus Fitness Association died. Notice the difference in the confidence intervals calculated in Example and the following Try It exercise. Why or why not? State the confidence interval. 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. Arrow down and enter the name of the list where the data is stored. 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. To receive certification from the Federal Communications Commission (FCC) for sale in the United States, the SAR level for a cell phone must be no more than 1.6 watts per kilogram. using a calculator, computer or a standard normal probability table. Construct a 90% confidence interval for the population mean, . Normal. Can we (with 75% confidence) conclude that at least half of all American adults believe this? Construct a 95% confidence interval for the population proportion of adult Americans who feel that crime is the main problem. Finding the standard deviation When we know the population standard deviation \(\sigma\), we use a standard normal distribution to calculate the error bound EBM and construct the confidence interval. Use the Student's \(t\)-distribution. The sample standard deviation is 2.8 inches. The stated \(\pm 3%\) represents the maximum error bound. 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? How should she explain the confidence interval to her audience? Mathematically, Suppose we have collected data from a sample. (This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the standard normal distribution. Find the point estimate for the population mean. "We estimate with ___% confidence that the true population mean (include the context of the problem) is between ___ and ___ (include appropriate units).". Construct a 90% confidence interval for the population mean number of letters campers send home. Table shows a different random sampling of 20 cell phone models. For example, when \(CL = 0.95, \alpha = 0.05\) and \(\dfrac{\alpha}{2} = 0.025\); we write \(z_{\dfrac{\alpha}{2}} = z_{0.025}\). Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. It is assumed that the distribution for the length of time they last is approximately normal. Summary: Effect of Changing the Sample Size. Construct a 90% confidence interval for the population proportion of people who feel the president is doing an acceptable job. A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes. Suppose that 14 children, who were learning to ride two-wheel bikes, were surveyed to determine how long they had to use training wheels. What will happen to the error bound and confidence interval if 500 community colleges were surveyed? 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. Suppose that our sample has a mean of \(\bar{x} = 10\), and we have constructed the 90% confidence interval (5, 15) where \(EBM = 5\). For example, if we constructed 100 of these confidence intervals, we would expect 90 of them to contain the true population mean exam score. To capture the true population mean, we need to have a larger interval. \(CL = 0.75\), so \(\alpha = 1 0.75 = 0.25\) and \(\frac{\alpha}{2} = 0.125 z_{\frac{\alpha}{2}} = 1.150\). Arrow to Stats and press ENTER. The sample mean is 15, and the error bound for the mean is 3.2. If we were to sample many groups of nine patients, 95% of the samples would contain the true population mean length of time. The standard deviation for this data to the nearest hundred is \(\sigma\) = $909,200. (5.87, 7.98) \(z = z_{0.025} = 1.96\), because the confidence level is 95%. So, to capture this uncertainty we can create a confidence interval that contains a range of values that are likely to contain the true mean weight of the turtles in the population. It will need to change the sample size. How many students must you interview? use the data and confidence level to construct a confidence interval estimate of p, then address the given question. This means that, for example, a 99% confidence interval will be wider than a 95% confidence interval for the same set of data. \(N 7.9\left(\frac{2.5}{\sqrt{20}}\right)\). Explain your choice. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site 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. Confidence Interval for a population mean - known Joshua Emmanuel 95.5K subscribers 467K views 6 years ago Normal Distribution, Confidence Interval, Hypothesis Testing This video shows. The FEC has reported financial information for 556 Leadership PACs that operating during the 20112012 election cycle. It was revealed that they used them an average of six months with a sample standard deviation of three months. 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. Construct and interpret a 90% confidence Do, Conclude) interval for mu = the true mean life span of Bulldogs. Note:You can also find these confidence intervals by using the Statology Confidence Interval Calculator. The Federal Election Commission (FEC) collects information about campaign contributions and disbursements for candidates and political committees each election cycle. (b) Construct the 90% confidence interval for the population mean if the sample size, n, is 25. We estimate with 96% confidence that the mean amount of money raised by all Leadership PACs during the 20112012 election cycle lies between $47,292.57 and $456,415.89. Confidence Intervals for (Known) Example : A random sample of 25 students had a grade point average with a mean of 2.86. The average height of young adult males has a normal distribution with standard deviation of 2.5 inches. Step 1: Our confidence level is 0.95 because we seek to create a 95% confidence interval. The CONFIDENCE function calculates the confidence interval for the mean of the population. The following data were collected: 20; 75; 50; 65; 30; 55; 40; 40; 30; 55; $1.50; 40; 65; 40. In words, define the random variable \(X\). How do you construct a 90% confidence interval for the population mean, ? The 90% confidence interval is (67.1775, 68.8225). \(N\left(23.6, \frac{7}{\sqrt{100}}\right)\) because we know sigma. Use a 90% confidence level. Stanford University conducted a study of whether running is healthy for men and women over age 50. We estimate with 93% confidence that the true SAR mean for the population of cell phones in the United States is between 0.8035 and 1.0765 watts per kilogram. Calculate the error bound based on the information provided. Construct a 99% confidence interval to estimate the population mean using the data below. Use the Student's t-distribution. Why? The formula for Confidence Interval can be calculated by using the following steps: Step 1: Firstly, determine the sample mean based on the sample observations from the population data set. To find the 98% confidence interval, find \(\bar{x} \pm EBM\). \(\bar{X}\) is the mean number of unoccupied seats from a sample of 225 flights. The error bound formula for an unknown population mean \(\mu\) when the population standard deviation \(\sigma\) is known is, \[EBM = z_{\alpha/2} \left(\dfrac{\sigma}{\sqrt{n}}\right)\nonumber \]. Construct a 92% confidence interval for the population mean number of unoccupied seats per flight. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study. 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? To find the confidence interval, start by finding the point estimate: the sample mean. Construct a 95% confidence interval for the population mean length of time. In Exercises 9-24, construct the confidence interval estimate of the mean. Remember to use the area to the LEFT of \(z_{\dfrac{\alpha}{2}}\); in this chapter the last two inputs in the invNorm command are 0, 1, because you are using a standard normal distribution \(Z \sim N(0, 1)\). One way to lower the sampling error is to increase the sample size. We are interested in the population proportion of drivers who claim they always buckle up. They randomly surveyed 400 drivers and found that 320 claimed they always buckle up. We will use a Students \(t\)-distribution, because we do not know the population standard deviation. 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. Use this sample data to construct a 90% confidence interval for the mean age of CEOs for these top small firms. Confidence levels are expressed as a percentage (for example, a 95% confidence level). How would the number of people the firm surveys change? If we don't know the sample mean: \(EBM = \dfrac{(68.8267.18)}{2} = 0.82\). c|net part of CBX Interactive Inc. A survey of 20 campers is taken. 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. percent of all Asians who would welcome a black person into their families. Find the error bound and the sample mean. And it says the population standard deviation is 15, so we actually have sigma here, the population standard deviation sigma is 15 and we're asked to find the 95% confidence interval for the mean amount spent per person per day at this particular um theme park. A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68. Assume that the population standard deviation is \(\sigma = 0.337\). That means that tn - 1 = 1.70. Confidence intervals are typically written as (some value) (a range). ), \(EBM = (1.96)\left(\dfrac{3}{\sqrt{36}}\right) = 0.98\). We are interested in the population proportion of people who feel the president is doing an acceptable job. Construct a 90% confidence interval to estimate the population mean using the data below. \(z_{\dfrac{\alpha}{2}} = z_{0.025} = 1.96\), when using invnorm(0.975,0,1) on the TI-83, 83+, or 84+ calculators. Assume that the underlying population distribution is normal. Assume the population has a normal distribution. B. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean. You need to interview at least 385 students to estimate the proportion to within 5% at 95% confidence. The formula for sample size is \(n = \dfrac{z^{2}\sigma^{2}}{EBM^{2}}\), found by solving the error bound formula for \(n\). Why? (Notice this is larger than the z *-value, which would be 1.96 for the same confidence interval.) 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. Use the original 90% confidence level. Did you expect it to be? Confidence Intervals. A camp director is interested in the mean number of letters each child sends during his or her camp session. Do you think that six packages of fruit snacks yield enough data to give accurate results? Past studies have shown that the standard deviation is 0.15 and the population is normally distributed. Explain why. { "7.01:_Confidence_Intervals" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.
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construct a 90% confidence interval for the population mean