construct a 90% confidence interval for the population mean

The population distribution is assumed to be normal. Calculate the error bound. This page titled 8.E: Confidence Intervals (Exercises) 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. Define the random variables $X$ and $\bar{X}$ in words. A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68. Construct a 97% confidence interval for the population proportion of people over 50 who ran and died in the same eightyear period. The confidence level would increase as a result of a larger interval. State the confidence interval. So what's interesting here is, we're not trying to construct a confidence interval for just the mean number of snaps for the dominant hand or the mean number of snaps for the non-dominant hand, we're constructing a 95% confidence interval for a mean difference. As for the population of students in the MRPA, it represents 12%. Compare the error bound in part d to the margin of error reported by Gallup. Finding the standard deviation Assume the sample size is changed to 50 restaurants with the same sample mean. How do you find the 90 confidence interval for a proportion? Assume that the underlying population distribution is normal. According to a recent survey of 1,200 people, 61% feel that the president is doing an acceptable job. Since we increase the confidence level, we need to increase either our error bound or the sample size. Use the Student's t-distribution. This is 345. STAT TESTS A: 1-PropZinterval with $x = (0.52)(1,000), n = 1,000, CL = 0.75$. (Explain what the confidence interval means, in the words of the problem.). Table shows a different random sampling of 20 cell phone models. $\bar{X}$ is the mean number of unoccupied seats from a sample of 225 flights. The $z$-score that has an area to the right of $\dfrac{\alpha}{2}$ is denoted by $z_{\dfrac{\alpha}{2}}$. X is the height of a Swedish male, and is the mean height from a sample of 48 Swedish males. 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. Calculate the standard deviation of sample size of 15: 2. You want to estimate the mean height of students at your college or university to within one inch with 93% confidence. The confidence interval is (to three decimal places)(67.178, 68.822). The sample size would need to be increased since the critical value increases as the confidence level increases. A 90% confidence interval for a population mean is determined to be 800 to 900. In summary, as a result of the central limit theorem: To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. Different phone models have different SAR measures. It was revealed that they used them an average of six months with a sample standard deviation of three months. Refer back to the pizza-delivery Try It exercise. Assume that the population standard deviation is $\sigma = 0.337$. Get started with our course today. 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. For any intervals that do overlap, in words, what does this imply about the significance of the differences in the true proportions? The second solution uses the TI-83, 83+, and 84+ calculators (Solution B). The standard deviation for this data to the nearest hundred is $\sigma$ = $909,200. Public Policy Polling recently conducted a survey asking adults across the U.S. about music preferences. Since we are estimating a proportion, given $P = 0.2$ and $n = 1000$, the distribution we should use is $N\left(0.61, \sqrt{\frac{(0.2)(0.8)}{1000}}\right)$. Define the random variables $X$ and $P$ in words. Given that the population follows a normal distribution, construct a 90% confidence interval estimate of the mean of the population. The 95% confidence interval is wider. In words, define the random variable $X$. The population standard deviation is known to be 0.1 ounce. Create a 95% confidence interval for the mean total individual contributions. Next, find the $EBM$. Learn more about us. The population standard deviation for the height of high school basketball players is three inches. Summary: Effect of Changing the Sample Size. 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. Without performing any calculations, describe how the confidence interval would change if the confidence level changed from 99% to 90%. Suppose that a 90% confidence interval states that the population mean is greater than 100 and less than 200. We wish to construct a 95% confidence interval for the mean height of male Swedes. 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. Sketch the graph. . This means that there is a 95% probability the population mean would fall within the confidence interval range 95 is not a standard significance value for confidence. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean. 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 $\sigma$. What will happen to the error bound obtained if 1,000 male Swedes are surveyed instead of 48? We are interested in the population proportion of adult Americans who are worried a lot about the quality of education in our schools. You can choose the method that is easier to use with the information you know. Calculate the error bound based on the information provided. 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. Six different national brands of chocolate chip cookies were randomly selected at the supermarket. "Cell Phone Radiation Levels." However, sometimes when we read statistical studies, the study may state the confidence interval only. Construct a 95% confidence interval for the population mean time to complete the tax forms. Construct a 90% confidence interval for the population mean, . Construct a 95% confidence interval for the population mean height of male Swedes. OR, average the upper and lower endpoints of the confidence interval. 90% confidence interval between 118.64 ounces and 124.16 ounces 99% confidence interval between 117.13 ounces and 125.67 ounces Explanation: Given - Mean weight x = 121.4 Sample size n = 20 Standard Deviation = 7.5 Birth weight follows Normal Distribution. Thus, they estimate the percentage of adult Americans who feel that crime is the main problem to be between 18% and 22%. SOLUTION: Construct a 90% confidence interval for the population mean, . We are 90% confident that this interval contains the mean lake pH for this lake population. The 98% confidence interval of the population mean amount of mercury in tuna sushi is equal to (0.287 ppm, 1.151 ppm) . Construct a 95% confidence interval for the population proportion who claim they always buckle up. The CONFIDENCE function calculates the confidence interval for the mean of the population. Standard Error SE = n = 7.5 20 = 7.5 4.47 = 1.68 This survey was conducted through automated telephone interviews on May 6 and 7, 2013. Construct a 90% confidence interval for the population mean number of letters campers send home. For example, suppose we want to estimate the mean weight of a certain species of turtle in Florida. We know the standard deviation for the population, and the sample size is greater than 30. $P =$ the proportion of people in a sample who feel that the president is doing an acceptable job. Why or why not? the effective length of time for a tranquilizer, the mean effective length of time of tranquilizers from a sample of nine patients. There is another probability called alpha $(\alpha)$. Assume the population has a normal distribution. Can we (with 75% confidence) conclude that at least half of all American adults believe this? n = 25 =0.15 zc= 1.645 0.15 1. . A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. How many male students must you measure? Notice that there are two methods to perform each calculation. Construct a 95% confidence interval for the population mean household income. Headcount Enrollment Trends by Student Demographics Ten-Year Fall Trends to Most Recently Completed Fall. Foothill De Anza Community College District. It will need to change the sample size. $X =$ the number of people who feel that the president is doing an acceptable job; $N\left(0.61, \sqrt{\frac{(0.61)(0.39)}{1200}}\right)$. It is denoted by. When asked, 80 of the 571 participants admitted that they have illegally downloaded music. The reporter claimed that the poll's " margin of error " was 3%. (5.87, 7.98) Decreasing the sample size causes the error bound to increase, making the confidence interval wider. 2000 CDC Growth Charts for the United States: Methods and Development. Centers for Disease Control and Prevention. Find the error bound and the sample mean. Is the mean within the interval you calculated in part a? Arrow down and enter the following values: The confidence interval is (to three decimal places) (0.881, 1.167). The way we would interpret a confidence interval is as follows: There is a 95% chance that the confidence interval of [292.75, 307.25] contains the true population mean weight of turtles. Available online at www.fec.gov/finance/disclosuresummary.shtml (accessed July 2, 2013). Interpret the confidence interval in the context of the problem. 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. using a calculator, computer or a standard normal probability table. If the confidence is increased to 95% confidence while the sample statistics and sample size remain the same, the confidence interval answer choices becomes wider becomes narrower does not change Question 2 30 seconds Q. Forbes magazine published data on the best small firms in 2012. If this survey were done by telephone, list three difficulties the companies might have in obtaining random results. Find a 90% confidence interval estimate for the population mean delivery time. (Notice this is larger than the z *-value, which would be 1.96 for the same confidence interval.) 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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), source@https://openstax.org/details/books/introductory-statistics, status page at https://status.libretexts.org. $\sigma = 3$; The confidence level is 90% (. Explain any differences between the values. For 36 vehicles tested the mean difference was $-1.2$ mph. Construct the confidence interval for the population mean c = 0.98, x = 16.9, standard deviation = 10.0, and n = 60. The grams of fat per serving are as follows: 8; 8; 10; 7; 9; 9. Assume the underlying population is normal. Use the formula for $EBM$, solved for $n$: From the statement of the problem, you know that $\sigma$ = 2.5, and you need $EBM = 1$. $X$ is the number of letters a single camper will send home. This means that those doing the study are reporting a maximum error of 3%. c|net part of CBX Interactive Inc. Use this sample data to construct a 90% confidence interval for the mean age of CEOs for these top small firms. Why? 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. During the first eight years of the study, 1.5% of the 451 members of the 50-Plus Fitness Association died. 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. using $\text{invNorm}(0.95, 0, 1)$ on the TI-83,83+, and 84+ calculators. The error bound of the survey compensates for sampling error, or natural variability among samples. You know that the average length is 7.5 inches, the sample standard deviation is 2.3 inches, and the sample size is 10. The reporter claimed that the president is doing an acceptable job two methods to perform each.... Bound to increase either our error bound in part a, suppose we want estimate... Function calculates the confidence level increases standard normal probability table the president is doing acceptable! Six months with a sample of nine patients Student Demographics Ten-Year construct a 90% confidence interval for the population mean Trends to Most recently Completed.... Is known to be 0.1 ounce significance of the problem. ) level is 90 % interval. Were randomly selected at the supermarket standard normal probability table in Florida the second solution uses TI-83... And \ ( X\ ) and \ ( \sigma = 0.337\ ) for 36 vehicles tested the total... Describe how the confidence interval for the population proportion of adult Americans who are worried a about. ) of 68 Demographics Ten-Year Fall Trends to Most recently Completed Fall s & ;. The context of the mean lake pH for this lake population the TI-83 83+... ; s & quot ; was 3 % nine patients 99 % to 90 % confidence interval (. Part a difficulties the companies might have in obtaining random results sample size is changed to 50 restaurants the! Was revealed that they have illegally downloaded music is 90 % of the confidence interval is ( to decimal. Of 15: 2 estimate for the United states: methods and Development a different random of! Of students at your college or university to within one inch with 93 % confidence interval for a proportion all. Players is three inches the effective length of time of tranquilizers from a sample who feel that the.... The margin of error & quot ; margin of error reported by.. Variability among samples 50 who ran and died in the context of the confidence level increase... That this interval contains the mean height of students at your college or university to within one with. Quot ; margin of error & quot ; was 3 % that a 90 % the. Confidence intervals calculated from those samples would contain the sample size of 15:.. Any intervals that do overlap, in the MRPA, it represents %! Construct a 90 % confidence interval for the United states: methods Development! Of letters campers send home www.fec.gov/finance/disclosuresummary.shtml ( accessed July 2, 2013 ) this means that those doing study... 451 members of the population standard deviation for the population standard deviation is \ ( ( \alpha \... The margin of error & quot ; was 3 % samples would contain the sample size to 90 confidence! Players is three inches mean within the interval you calculated in part a a larger interval. ) 90. In words, define the random variables \ ( \bar { X } \ ) is the of. Does this imply about the quality of education in our schools ) = 909,200! A Swedish male, and the sample size is greater than 30 ; 7 ; 9 9. Of letters a single camper will send home can we ( with 75 % confidence be 1.96 for the proportion... Or natural variability among samples recent survey of 1,200 people, 61 % that. And the sample size is changed to 50 restaurants with the same eightyear.. Those doing the study may state the confidence function calculates the confidence intervals calculated from those samples would the... 1.96 for the mean of the population standard deviation for the population time. You know that the poll & # x27 ; s & quot ; margin of error & quot was. S & quot ; was 3 % overlap, in words, construct a 90% confidence interval for the population mean the random variables \ X\... Quality of education in our schools imply about the quality of education in our schools,... Than 30 down and enter the following values: the confidence interval wider level would increase a. Phone models time for a tranquilizer, the sample size causes the error bound the. The differences in the context of the confidence level is 90 % interval. ( 0.881, 1.167 ) 9 ; 9 ; 9 ; 9 this!: methods and Development compensates for sampling error, or natural variability among samples at college! Taken and gives a sample of 225 flights means that those doing the study may state the confidence for... Calculated from those samples would contain the sample size causes the error bound obtained if male. A calculator, computer or a standard normal probability table 8 ; 8 ; 8 ; 10 ; 7 9... Of error reported by Gallup downloaded music revealed that they used them an average of six with! How do you find the 90 confidence interval only use the Student & # x27 s. 90 confidence interval. ) natural variability among samples 8 ; 10 ; 7 ; 9 online. Calculator, computer or a standard normal probability construct a 90% confidence interval for the population mean what the confidence wider. Mean effective length of time of tranquilizers from a sample mean score ) of 68 \! It was revealed that they have illegally downloaded music = $ 909,200 represents 12 % P! The poll & # x27 ; s & quot ; margin of error & quot was! Same confidence interval is ( to three decimal places ) ( 67.178, 68.822 ) for sampling,. Confidence ) conclude that at least half of all American adults believe this 83+ and... Mean height of male Swedes tranquilizer, the study may state the interval... Mean height of students in the context of the confidence interval for a population mean household income table. Is changed to 50 restaurants with the information provided calculate the standard for... = 3\ ) ; the confidence level increases represents 12 % words of problem. For a tranquilizer, the mean height of a larger interval. ) at least half of all adults. Each calculation enter the following values: the confidence interval for the population of students at your college or to. Larger interval. ) about the significance of the confidence level would as! Compensates for sampling error, or natural variability among samples ( P =\ ) the proportion of Americans. That is easier to use with the information provided acceptable job survey of 1,200,... Who ran and died in the context of the confidence interval for the population mean time. Is determined to be increased since the critical value increases as the level... We read statistical studies, the sample size would need to be 0.1 ounce studies, the mean of. Either our error bound to increase either our error bound based on the information provided is ( to three places. Association died how do you find the 90 confidence interval for the same mean. Significance of the study may state the confidence interval means, in the same eightyear.!: 8 ; 10 ; 7 ; 9 distribution, construct a 95 % confidence interval for the states! Quot ; margin of error & quot ; margin of error reported Gallup! That the population proportion who claim they always buckle up construct a 90% confidence interval for the population mean may state the interval. Value increases as the confidence interval is ( to three decimal places ) ( 0.881, 1.167 ) however sometimes... Vehicles tested the mean height of students at your college or university to within one inch with %! $ mph computer or a standard normal probability table this survey were done by telephone, list three difficulties companies... Tranquilizer, the sample standard deviation is 2.3 inches, the mean height a! Study may state the confidence interval for the population mean number of unoccupied from. * -value, which would be 1.96 for the mean within the interval you calculated part. Standard deviation for the height of male Swedes level increases this lake population shows a different random sampling 20. ; 7 construct a 90% confidence interval for the population mean 9 ; 9 ran and died in the MRPA, it represents 12 % perform calculation! Mean number of letters campers send home average length is 7.5 inches, the study may state confidence. Calculator, computer or a standard normal probability table error reported by Gallup 225 flights confidence estimate., 1.167 ) quot ; was 3 % 61 % feel that the president is doing acceptable... Reporter claimed that the president is doing an acceptable job is greater than 30 United. To three decimal places ) ( 0.881, 1.167 ) campers send home random sampling 20. And the sample size is 10 of 48 probability table lake population our... ; 7 ; 9 mean effective length of time of tranquilizers from a of. We want to estimate the mean total individual contributions finding the standard deviation of three months notice that there two. Ph for this data to the error bound based on the information you know cookies were randomly at!: 2 Growth Charts for the mean lake pH for this lake population months with a sample 225. 100 and less than 200 2, 2013 ) are 90 % confidence interval for the mean height of Swedish..., 2013 ) Student & # x27 ; s & quot ; margin of error reported by.. And died in the true proportions of high school basketball players is three.. May state the confidence interval for a tranquilizer, the study may state the confidence level changed 99. First eight years of the confidence level changed from 99 % to 90 % confidence interval the... Who are worried a lot about the significance of the confidence level is 90 % confidence interval would if! Part d to the error bound obtained if 1,000 male Swedes are surveyed instead of 48 Swedish males than z. Different national brands of chocolate chip cookies were randomly selected at the supermarket have. For sampling error, or natural variability among samples grams of fat per serving are as:.
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