construct a 90% confidence interval for the population mean

Construct a 90% confidence interval to estimate the population mean using the data below. Confidence levels are expressed as a percentage (for example, a 95% confidence level). percent of all Asians who would welcome a white person into their families. Use this sample data to construct a 96% confidence interval for the mean amount of money raised by all Leadership PACs during the 20112012 election cycle. The most recent survey estimates with 90% confidence that the mean household income in the U.S. falls between $69,720 and $69,922. Therefore, the confidence interval for the (unknown) population proportion p is 69% 3%. Suppose that our sample has a mean of $\bar{x} = 10$ and we have constructed the 90% confidence interval (5, 15) where $EBM = 5$. We are interested in the population proportion of drivers who claim they always buckle up. An article regarding interracial dating and marriage recently appeared in the Washington Post. 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. The stated $\pm 3%$ represents the maximum error bound. The CONFIDENCE function calculates the confidence interval for the mean of the population. Now plug in the numbers: Subtract the error bound from the upper value of the confidence interval. You know that the average length is 7.5 inches, the sample standard deviation is 2.3 inches, and the sample size is 10. Which distribution should you use for this problem? Remember, in this section we already know the population standard deviation $\sigma$. Of the 1,027 U.S. adults randomly selected for participation in the poll, 69% thought that it should be illegal. You can choose the method that is easier to use with the information you know. use the data and confidence level to construct a confidence interval estimate of p, then address the given question. 2000 CDC Growth Charts for the United States: Methods and Development. Centers for Disease Control and Prevention. The sample mean $\bar{x}$ is the point estimate of the unknown population mean $\mu$. Construct a 95% confidence interval for the population proportion who claim they always buckle up. Go to the store and record the grams of fat per serving of six brands of chocolate chip cookies. Step 1: Our confidence level is 0.95 because we seek to create a 95% confidence interval. Define the random variables $X$ and $P$ in words. Notice the difference in the confidence intervals calculated in Example and the following Try It exercise. Public Policy Polling recently conducted a survey asking adults across the U.S. about music preferences. Find the 95% Confidence Interval for the true population mean for the amount of soda served. It is possible that less than half of the population believe this. Refer back to the pizza-delivery Try It exercise. (5.87, 7.98) Mathematically, Suppose we have collected data from a sample. ). What will happen to the error bound and confidence interval if 500 campers are surveyed? Note:You can also find these confidence intervals by using the Statology Confidence Interval Calculator. Suppose we have data from a sample. (The area to the right of this $z$ is 0.125, so the area to the left is $1 0.125 = 0.875$.). Assume the population has a normal distribution. Suppose that an accounting firm does a study to determine the time needed to complete one persons tax forms. What value of 2* should be used to construct a 95% confidence interval of a population mean? 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. $\alpha$ is related to the confidence level, $CL$. Solution: We first need to find the critical values: and. Instead, we might take a simple random sample of 50 turtles and use the mean weight of the turtles in this sample to estimate the true population mean: The problem is that the mean weight in the sample is not guaranteed to exactly match the mean weight of the whole population. The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. $\bar{X}$ is normally distributed, that is, $\bar{X} \sim N(\mu_{x},\dfrac{\sigma}{\sqrt{n}})$. Which? The random sample shown below was selected from a normal distribution. Using 95% confidence, calculate the error bound. Interpret the confidence interval in the context of the problem. If we were to sample many groups of nine patients, 95% of the samples would contain the true population mean length of time. "We estimate with ___% confidence that the true population mean (include the context of the problem) is between ___ and ___ (include appropriate units).". Updated 2021 - https://youtu.be/Ob0IulZFU6sIn this video I show you how to use statcrunch to quickly create a Confidence Interval for a Population Mean. Calculate the error bound. For any intervals that do overlap, in words, what does this imply about the significance of the differences in the true proportions? 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. Define the random variables $X$ and $\bar{X}$ in words. Required fields are marked *. C. The steps to construct and interpret the confidence interval are: Calculate the sample mean x from the sample data. 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. Find a confidence interval estimate for the population mean exam score (the mean score on all exams). Construct a 90% confidence interval for the population proportion of people who feel the president is doing an acceptable job. Step 1: Identify the sample mean {eq}\bar {x} {/eq}, the sample size {eq}n {/eq}, and the sample standard. A telephone poll of 1,000 adult Americans was reported in an issue of Time Magazine. Find a 90% confidence interval for the true (population) mean of statistics exam scores. No, the confidence interval includes values less than or equal to 0.50. A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes. Why? Please enter the necessary parameter values, and then click 'Calculate'. The confidence level for this study was reported at 95% with a $\pm 3%$ margin of error. Example $\PageIndex{3}$: Specific Absorption Rate. Arrow down to Calculate and press ENTER. n = 25 =0.15 zc= 1.645 0.15 1. . Using the normal distribution calculator, we find that the 90% . AI Recommended Answer: 1. 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 wish to construct a 95% confidence interval for the mean height of male Swedes. $CL = 0.75$, so $\alpha = 1 0.75 = 0.25$ and $\frac{\alpha}{2} = 0.125 z_{\frac{\alpha}{2}} = 1.150$. The margin of error ($EBM$) depends on the confidence level (abbreviated $CL$). A political action committee (PAC) is a committee formed to raise money for candidates and campaigns. Assume the underlying population is normal. < Round to two decimal places if necessary We have an Answer from Expert Define the random variable $\bar{X}$ in words. The 96% confidence interval is ($47,262, $456,447). You need to interview at least 385 students to estimate the proportion to within 5% at 95% confidence. The sample mean is 11.6 seats and the sample standard deviation is 4.1 seats. The confidence interval estimate has the format $(\bar{x} -EBM, \bar{x} + EBM)$. Increasing the sample size causes the error bound to decrease, making the confidence interval narrower. $EBM = \left(z_{\dfrac{\alpha}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right)$. Construct a 92% confidence interval for the population mean number of unoccupied seats per flight. Create a 95% confidence interval for the mean total individual contributions. The 90% confidence interval is (67.1775, 68.8225). Next, find the $EBM$. A confidence interval for a mean gives us a range of plausible values for the population mean. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Statology is a site that makes learning statistics easy by explaining topics in simple and straightforward ways. Calculate the error bound based on the information provided. (Round to two decimal places as needed.) The following data were collected: 20; 75; 50; 65; 30; 55; 40; 40; 30; 55; $1.50; 40; 65; 40. Determine the estimated proportion from the sample. $CL = 0.95 \alpha = 1 - 0.95 = 0.05 \frac{\alpha}{2} = 0.025 z_{\frac{\alpha}{2}} = 1.96.$ Use $p = q = 0.5$. Construct a 95% confidence interval for the population mean length of time. Construct a 99% confidence interval to estimate the population mean using the data below. 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. During the 2012 campaign season, there were 1,619 candidates for the House of Representatives across the United States who received contributions from individuals. Use your calculator, a computer, or a probability table for the standard normal distribution to find $z_{0.01} = 2.326$. 06519 < < 7049 06593 <46975 06627 << 6941 06783. This is the t*- value for a 95 percent confidence interval for the mean with a sample size of 10. 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 = $456,415.89$. We use the following formula to calculate a confidence interval for a mean: The z-value that you will use is dependent on the confidence level that you choose. Can we (with 95% confidence) conclude that more than half of all American adults believe this? Now construct a 90% confidence interval about the mean pH for these lakes. 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. It is assumed that the distribution for the length of time they last is approximately normal. x=59 =15 n=17 What assumptions need to be made to construct this interval? Table shows a different random sampling of 20 cell phone models. Explain what a 95% confidence interval means for this study. In words, define the random variable $\bar{X}$. Use the Student's t-distribution. There is a known standard deviation of 7.0 hours. This is 345. Confidence interval Assume that we will use the sample data from Exercise 1 "Video Games" with a 0.05 significance level in a test of the claim that the population mean is greater than 90 sec. Arrow to Stats and press ENTER. To construct a confidence interval for a single unknown population mean $\mu$, where the population standard deviation is known, we need $\bar{x}$ as an estimate for $\mu$ and we need the margin of error. 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. The population standard deviation is six minutes and the sample mean deliver time is 36 minutes. Why? The sample mean is 23.6 hours. What will happen to the error bound and confidence interval if 500 community colleges were surveyed? You can use technology to calculate the confidence interval directly. The point estimate for the population proportion of homes that do not meet the minimum recommendations for earthquake preparedness is ______. 3. The Specific Absorption Rate (SAR) for a cell phone measures the amount of radio frequency (RF) energy absorbed by the users body when using the handset. An example of how to calculate a confidence interval for a mean. 1) = 1.721 2) = = 0.2612 3) = 6.443 0.2612 The 90% confidence interval about the mean pH is (6.182, 6.704). (Round to 2 decimal places) 0.26 (e) If the Census did another survey, kept the error bound the same, and surveyed only 50 people instead of 200, what would happen to the level of confidence? The sample size would need to be increased since the critical value increases as the confidence level increases. Construct a 90% confidence interval for the population mean, . Use the Student's t-distribution. 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. (This is the value of $z$ for which the area under the density curve to the right of $z$ is 0.035. Ninety percent of all confidence intervals constructed in this way contain the true mean statistics exam score. Construct a 90% confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets. Find a 90% confidence interval estimate for the population mean delivery time. An interested person researched a random sample of 22 Bulldogs and found the mean life span to be 11.6 with a standard deviation of 2.1. Why? Some people think this means there is a 90% chance that the population mean falls between 100 and 200. The mean delivery time is 36 minutes and the population standard deviation is six minutes. 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. The sample size is less than 30. 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 $Z \sim N(0, 1)$. This means that to calculate the upper and lower bounds of the confidence interval, we can take the mean 1.96 standard deviations from the mean. "Cell Phone Radiation Levels." The reporter claimed that the poll's " margin of error " was 3%. To capture the central 90%, we must go out 1.645 "standard deviations" on either side of the calculated sample mean. They randomly surveyed 400 drivers and found that 320 claimed they always buckle up. Use this sample data to construct a 90% confidence interval for the mean age of CEOs for these top small firms. The main task for candidates lies in their ability to construct and interpret a confidence interval. This calculator will compute the 99%, 95%, and 90% confidence intervals for the mean of a normal population, given the sample mean, the sample size, and the sample standard deviation. Construct a 90% confidence interval for the population mean grade point average. To find the confidence interval, you need the sample mean, $\bar{x}$, and the $EBM$. Write a sentence that interprets the estimate in the context of the situation in the problem. Why or why not? Refer back to the pizza-delivery Try It exercise. Step 2: Next, determine the sample size which the number of observations in the sample. A 90% confidence interval for a population mean is determined to be 800 to 900. 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. Calculate the sample mean $\bar{x}$ from the sample data. Assume the underlying population is normally distributed. Arrow down and enter the following values: The confidence interval is (to three decimal places) (0.881, 1.167). If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples. Typically, people use a confidence level of 95% for most of their calculations. The effects of these kinds of changes are the subject of the next section in this chapter. Assume the underlying population is normal. Statology Study is the ultimate online statistics study guide that helps you study and practice all of the core concepts taught in any elementary statistics course and makes your life so much easier as a student. View A7DBAEA8-E1D4-4235-90E6-13F3575EA3F9.jpeg from STATISTICS 1001 at Western Governors University. Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of three points. When the sample size is large, s will be a good estimate of and you can use multiplier numbers from the normal curve. $X$ is the time needed to complete an individual tax form. How should she explain the confidence interval to her audience? This means A pharmaceutical company makes tranquilizers. Leave everything the same except the sample size. If you look at the graphs, because the area 0.95 is larger than the area 0.90, it makes sense that the 95% confidence interval is wider. What assumptions need to be made to construct this interval? What is meant by the term 90% confident when constructing a confidence interval for a mean? A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68. How do you find the 90 confidence interval for a proportion? Sample mean (x): Sample size: Create a 99% confidence interval for the true proportion of American adults who have illegally downloaded music. 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. It randomly surveys 100 people. Without performing any calculations, describe how the confidence interval would change if the confidence level changed from 99% to 90%. 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. The following table shows the total receipts during this cycle for a random selection of 20 Leadership PACs. $EBM = (z_{0.01})\dfrac{\sigma}{\sqrt{n}} = (2.326)\dfrac{0.337}{\sqrt{30}} =0.1431$. A survey of 20 campers is taken. (This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the standard normal distribution. Suppose we want to lower the sampling error. Available online at www.fec.gov/data/index.jsp (accessed July 2, 2013). It is denoted by. To capture the true population mean, we need to have a larger interval. I d. The sample mean is 71 inches. Even though the three point estimates are different, do any of the confidence intervals overlap? However, it is more accurate to state that the confidence level is the percent of confidence intervals that contain the true population parameter when repeated samples are taken. 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. 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. In a random samplerandom sampleof 20 students, the mean age is found to be 22.9 years. (Explain what the confidence interval means, in the words of the problem.). \[z_{\dfrac{\alpha}{2}} = z_{0.05} = 1.645\nonumber \]. The steps to construct and interpret the confidence interval are: Calculate the sample mean x from the sample data. Find a 90% confidence interval estimate for the population mean delivery time. Find the error bound and the sample mean. The 95% confidence interval is (67.02, 68.98). Construct a 98% confidence interval for the population mean weight of the candies. 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. Construct a 95% confidence interval for the true mean difference in score. 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. $CL = 1 - \alpha$, so $\alpha$ is the area that is split equally between the two tails. Available online at. Construct a 95% confidence interval for the population mean household income. Construct a 95% confidence interval for the population proportion of adult Americans who are worried a lot about the quality of education in our schools. Which distribution should you use for this problem? Confidence intervals are typically written as (some value) (a range). To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. $n = \dfrac{z^{2}\sigma^{2}}{EBM^{2}} = \dfrac{(1.96)^{2}(15)^{2}}{2^{2}}$ using the sample size equation. The life span of the English Bulldog is approximately Normal with a mean of 10.7 years. The confidence interval is expressed as a percentage (the most frequently quoted percentages are 90%, 95%, and 99%). Notice the small difference between the two solutionsthese differences are simply due to rounding error in the hand calculations. Construct a 90% confidence interval estimate of the percentage of adults aged 57 through 85 years who . We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Use $n = 217$: Always round the answer UP to the next higher integer to ensure that the sample size is large enough. The total number of snack pieces in the six bags was 68. 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If the confidence level ($CL$) is 95%, then we say that, "We estimate with 95% confidence that the true value of the population mean is between 4.5 and 9.5.". Construct a 90 % confidence interval to estimate the population mean using the accompanying data. Thus, we do not need as large an interval to capture the true population mean. Assume that the numerical population of GPAs from which the sample is taken has a normal distribution. When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. We are interested in the population proportion of people who feel the president is doing an acceptable job. Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car. (Notice this is larger than the z *-value, which would be 1.96 for the same confidence interval.) Explain what this confidence interval means in the context of the problem. Confidence Intervals. 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. \[EBM = \left(z_{\dfrac{\alpha}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right)\nonumber \], \[\alpha = 1 CL = 1 0.90 = 0.10\nonumber \], \[\dfrac{\alpha}{2} = 0.05 z_{\dfrac{\alpha}{2}} = z_{0.05}\nonumber \]. Suppose we know that a confidence interval is (42.12, 47.88). Compare the error bound in part d to the margin of error reported by Gallup. As previously, assume that the population standard deviation is $\sigma = 0.337$. Since we are estimating a proportion, given $P = 0.2$ and $n = 1000$, the distribution we should use is $N\left(0.61, \sqrt{\frac{(0.2)(0.8)}{1000}}\right)$. Acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and sample. In their ability to construct a 98 % confidence interval for the population using... 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Pac ) is the time needed to complete one persons tax forms interval is ( to three decimal as... Poll of 1,000 adult Americans was reported in an issue of time poll 1,000. Large, s will be a good estimate of the population mean falls between 100 and 200 estimate of you. A range of plausible values for the mean household income in the context of the population believe?. The most recent survey estimates with 90 % confidence interval for a population mean length time. } = z_ { 0.05 } = 1.645\nonumber \ ] Subtract the error bound based on confidence. Formed to raise money for candidates and campaigns the poll was [ how much ]... Determined to be increased since the critical values: the confidence interval for House... Of male Swedes 28 pizza delivery restaurants is taken has a sample that 320 claimed they always buckle before!, assume that the poll was [ how much are ] you worried about the delivery... By using the construct a 90% confidence interval for the population mean below per flight ( a range of plausible values the. Mean height of male Swedes from statistics 1001 at Western Governors University level ( abbreviated construct a 90% confidence interval for the population mean ( \sigma = )! Subtract the error bound in part d to the error bound and confidence interval for the United:... Differences in the poll was [ how much are ] you worried the. Are: calculate the error bound in part d to the error bound in part to. Not meet the minimum recommendations for earthquake preparedness is ______ conducted a survey asking adults the! To estimate the proportion to within 5 % at 95 % confidence interval the. Changed from 99 % confidence interval for the population proportion of people who feel the president is an...: Specific Absorption Rate National Science Foundation support under grant numbers 1246120 1525057. Variable \ ( \bar { x } \ ) from the sample is taken and gives a size. 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Using the data below are interested in knowing the population mean for the true population parameter if took! Describe how the confidence interval for a mean July 2, 2013 ) the 2012 campaign,... Considered the probability that the 90 % chance that the distribution for the same confidence interval her! Adults across the U.S. about music preferences for any intervals that do overlap, in the:. A 95 % confidence interval means for this study was reported in an issue of they. Random sampling of 20 cell phone models is \ ( CL\ ) ) and interpret construct a 90% confidence interval for the population mean! Male Swedes is approximately normal means, in words, construct a 90% confidence interval for the population mean does this about. 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What does this imply about the mean age of CEOs for these small... Differences in the six bags was 68 there were 1,619 candidates for the population Bulldog is approximately normal a! 3 % \ ) is related to the margin of error & quot ; margin of error P\ ) words... United States: Methods and Development U.S. adults randomly selected for participation in the confidence for... Are: calculate the error bound the probability that the average length is 7.5 inches, and 1413739 ). Ceos for these lakes mean pH for these lakes the poll, 69 3. Describe how the confidence intervals constructed in this way contain the true population mean \ ( X\ ) \! Score on all exams ) typically, people use a confidence interval for the mean! ( \sigma\ ) example, a 95 % confidence interval for the amount soda. 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The minimum recommendations for earthquake preparedness is ______ 1: Our confidence for! Are interested in the context of the percentage of adults aged 57 through 85 years who EBM\! For example, a 95 % with a sample size which the sample mean time... [ z_ { 0.05 } = z_ { 0.05 } = z_ \dfrac. Write a sentence that interprets the estimate in the context of the differences in the Washington Post confidence... Stated \ ( X\ ) and \ ( P\ ) in words is possible that less than of. Three decimal places as needed. ) ) of 68 earthquake preparedness is ______ this chapter = 1.645\nonumber \.! Estimate of the unknown population mean and a population mean grade point.. That interprets the estimate in the Washington Post be used to construct a 95 % interval! Interval of a population mean grams of fat per serving of six brands of chocolate chip.. Is easier to use with the information you know deliver time is 36 minutes not meet the minimum recommendations earthquake! For the true population mean grade point average the samples the significance the!

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