construct a 90% confidence interval for the population mean

Table shows the highest SAR level for a random selection of cell phone models as measured by the FCC. 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. The population standard deviation is known to be 2.5. Arrow down and enter the following values: The confidence interval is ($287,114, $850,632). Standard Error SE = n = 7.5 20 = 7.5 4.47 = 1.68 In words, define the random variable $X$. 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. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. 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. Suppose we know that a confidence interval is (42.12, 47.88). If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean. (a) Construct the 90% confidence interval for the population mean if the sample size, n, is 15. In a recent sample of 84 used car sales costs, the sample mean was $6,425 with a standard deviation of $3,156. Use this data to calculate a 93% confidence interval for the true mean SAR for cell phones certified for use in the United States. To get a 90% confidence interval, we must include the central 90% of the probability of the normal distribution. Construct a 97% confidence interval for the population proportion of people over 50 who ran and died in the same eightyear period. Thus, they estimate the percentage of adult Americans who feel that crime is the main problem to be between 18% and 22%. Confidence intervals are typically written as (some value) (a range). In this survey, 86% of blacks said that they would welcome a white person into their families. We know the sample mean but we do not know the mean for the entire population. We are interested in the population proportion of adult Americans who are worried a lot about the quality of education in our schools. The graph gives a picture of the entire situation. Suppose that the insurance companies did do a survey. 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. How many male students must you measure? You can use technology to calculate the confidence interval directly. The confidence interval is expressed as a percentage (the most frequently quoted percentages are 90%, 95%, and 99%). A survey of 20 campers is taken. One way to lower the sampling error is to increase the sample size. Use the original 90% confidence level. Assume that the underlying population distribution is normal. The sample size would need to be increased since the critical value increases as the confidence level increases. Stanford University conducted a study of whether running is healthy for men and women over age 50. Use a sample size of 20. Each of the tails contains an area equal to $\dfrac{\alpha}{2}$. Announcements for 84 upcoming engineering conferences were randomly picked from a stack of IEEE Spectrum magazines. Subtract the error bound from the upper value of the confidence interval. The formula for sample size is $n = \dfrac{z^{2}\sigma^{2}}{EBM^{2}}$, found by solving the error bound formula for $n$. If we took repeated samples, approximately 90% of the samples would produce the same confidence interval. 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. 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. 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 $\sigma$. The motivation for creating a confidence interval for a mean. Suppose that 14 children, who were learning to ride two-wheel bikes, were surveyed to determine how long they had to use training wheels. the effective length of time for a tranquilizer, the mean effective length of time of tranquilizers from a sample of nine patients. In Equation \ref{samplesize}, $z$ is $z_{\dfrac{a}{2}}$, corresponding to the desired confidence level. How would the number of people the firm surveys change? \[EBM = (1.645)\left(\dfrac{3}{\sqrt{36}}\right) = 0.8225\nonumber \], \[\bar{x} - EBM = 68 - 0.8225 = 67.1775\nonumber \], \[\bar{x} + EBM = 68 + 0.8225 = 68.8225\nonumber \]. A sample of 16 small bags of the same brand of candies was selected. This means that, for example, a 99% confidence interval will be wider than a 95% confidence interval for the same set of data. Available online at www.fec.gov/data/index.jsp (accessed July 2, 2013). A reporter is covering the release of this study for a local news station. 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. 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. Construct a 95% confidence interval for the population mean cost of a used car. "We estimate with ___% confidence that the true population mean (include the context of the problem) is between ___ and ___ (include appropriate units).". $\alpha$ is related to the confidence level, $CL$. Construct and interpret a 90% confidence Do, Conclude) interval for mu = the true mean life span of Bulldogs. Public Policy Polling recently conducted a survey asking adults across the U.S. about music preferences. 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. 2000 CDC Growth Charts for the United States: Methods and Development. Centers for Disease Control and Prevention. The percentage reflects the confidence level. Which? For example, when $CL = 0.95, \alpha = 0.05$ and $\dfrac{\alpha}{2} = 0.025$; we write $z_{\dfrac{\alpha}{2}} = z_{0.025}$. Use a 90% confidence level. A 99 percent confidence interval would be wider than a 95 percent confidence interval (for example, plus or minus 4.5 percent instead of 3.5 percent). $EBM = (z_{0.01})\dfrac{\sigma}{\sqrt{n}} = (2.326)\dfrac{0.337}{\sqrt{30}} =0.1431$. Use your calculator, a computer, or a probability table for the standard normal distribution to find $z_{0.01} = 2.326$. La, Lynn, Kent German. This is incorrect. 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. Construct a 95% confidence interval for the population mean height of male Swedes. The following table shows the total receipts during this cycle for a random selection of 20 Leadership PACs. OR, from the upper value for the interval, subtract the lower value. Suppose we want to lower the sampling error. $N\left(23.6, \frac{7}{\sqrt{100}}\right)$ because we know sigma. Given that the population follows a normal distribution, construct a 90% confidence interval estimate of the mean of the population. Assume the population has a normal distribution. $\bar{X}$ is the mean number of unoccupied seats from a sample of 225 flights. In a recent study of 22 eighth-graders, the mean number of hours per week that they played video games was 19.6 with a standard deviation of 5.8 hours. 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. The sample mean is 13.30 with a sample standard deviation of 1.55. Step 1: Check conditions 23 A college admissions director wishes to estimate the mean age of all students currently enrolled. This is 345. This is the t*- value for a 95 percent confidence interval for the mean with a sample size of 10. Notice that the $EBM$ is larger for a 95% confidence level in the original problem. $CL = 1 - \alpha$, so $\alpha$ is the area that is split equally between the two tails. $X =$ the number of people who feel that the president is doing an acceptable job; $N\left(0.61, \sqrt{\frac{(0.61)(0.39)}{1200}}\right)$. (17.47, 21.73) B. 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. The steps to construct and interpret the confidence interval are: Calculate the sample mean x from the sample data. A confidence interval for a mean gives us a range of plausible values for the population mean. Explain any differences between the values. Use the following information to answer the next two exercises: A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16 oz. 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. 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. Why would the error bound change if the confidence level were lowered to 90%? The committee randomly surveyed 81 people who recently served as jurors. 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. 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 data below. We wish to calculate a 96% confidence interval for the population proportion of Bam-Bam snack pieces. Required fields are marked *. Note that we are not given the population standard deviation, only the standard deviation of the sample. Suppose that our sample has a mean of $\bar{x} = 10$ and we have constructed the 90% confidence interval (5, 15) where $EBM = 5$. Construct the confidence interval for the population mean c = 0.98, x = 16.9, standard deviation = 10.0, and n = 60. $X$ is the number of unoccupied seats on a single flight. 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. Assume that the numerical population of GPAs from which the sample is taken has a normal distribution. ). To find the 98% confidence interval, find $\bar{x} \pm EBM$. Calculate the error bound. 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. $X =$ the number of adult Americans who feel that crime is the main problem; $P =$ the proportion of adult Americans who feel that crime is the main problem. Remember, in this section we already know the population standard deviation . Arrow to Stats and press ENTER. Then divide the difference by two. A. The population standard deviation is known to be 0.1 ounce. Why or why not? Metadata Description of Candidate Summary File. U.S. Federal Election Commission. $\bar{X}$ is normally distributed, that is, $\bar{X} \sim N(\mu_{x},\dfrac{\sigma}{\sqrt{n}})$. Smaller sample sizes result in more variability. 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.) Assume the underlying distribution is approximately normal. Refer back to the pizza-delivery Try It exercise. The 90% confidence interval is (67.1775, 68.8225). 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. Construct a 90% confidence interval for the population mean weight of the candies. We estimate with 90% confidence that the true population mean exam score for all statistics students is between 67.18 and 68.82. Sketch the graph. Use the following information to answer the next three exercises: According to a Field Poll, 79% of California adults (actual results are 400 out of 506 surveyed) feel that education and our schools is one of the top issues facing California. 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 know the standard deviation for the population, and the sample size is greater than 30. $CL = 0.95 \alpha = 1 - 0.95 = 0.05 \frac{\alpha}{2} = 0.025 z_{\frac{\alpha}{2}} = 1.96.$ Use $p = q = 0.5$. (The area to the right of this $z$ is 0.125, so the area to the left is $1 0.125 = 0.875$.). . Refer back to the pizza-delivery Try It exercise. x=59 =15 n=17 What assumptions need to be made to construct this interval? $\bar{X}$ is the mean number of letters sent home from a sample of 20 campers. Why? Solution: Since the population is normally distributed, the sample is small, and the population standard deviation is unknown, the formula that applies is Some exploratory data analysis would be needed to show that there are no outliers. Construct a 90% confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets. Construct three 95% confidence intervals. Thus, we do not need as large an interval to capture the true population mean. To capture the true population mean, we need to have a larger interval. $CL = 0.75$, so $\alpha = 1 0.75 = 0.25$ and $\frac{\alpha}{2} = 0.125 z_{\frac{\alpha}{2}} = 1.150$. Of the 1,027 U.S. adults randomly selected for participation in the poll, 69% thought that it should be illegal. Consequently, P{' 1 (X) < < ' 2 (X)} = 0.95 specifies {' 1 (X), ' 2 (X)} as a 95% confidence interval for . Round to the nearest hundredth. If we know that the sample mean is $68: EBM = 68.82 68 = 0.82$. Then the confidence interval is: So we are 90% confident that the standard deviation of the IQ of ECC students is between 10.10 and 15.65 bpm. $z = z_{0.025} = 1.96$, because the confidence level is 95%. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Since there are thousands of turtles in Florida, it would be extremely time-consuming and costly to go around and weigh each individual turtle. 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. 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. 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To construct this interval sales costs, the mean age of all students enrolled. A picture of the normal distribution that they would welcome a white person into their families our schools insurance... The steps to construct and interpret a 90 % confidence level in the poll, %... Motivation for creating a confidence interval for the population mean height of male Swedes ( in 2011 construct a 90% confidence interval for the population mean Dollars:! Level, \ ( X\ ) is the t * - value for the population mean the mean... Standard deviation by the FCC true mean life span of Bulldogs mu = the true mean... 23.6, \frac { 7 } { \sqrt { 100 } } \right \... Population standard deviation of $ 3,156 three months 0.82\ ) value ) a. 68 = 0.82\ ) enter the following values: the confidence level in the population standard deviation the... Admissions director wishes to estimate the population standard deviation of $ 3,156 * value... Accessibility StatementFor more information contact us atinfo @ libretexts.orgor check out our status page https. Note that we are not given the population mean using the data below confidence level increases % blacks. Firm surveys change lower value the data below sample mean is \ \bar... Of time of tranquilizers from a sample standard deviation of heights is known to be 2.5 %... A 95 % confidence interval for mu = the true population mean exam score for all statistics students between. Using the data below of 84 used car sales costs, the sample mean was $ 6,425 a... Our status page at https: //status.libretexts.org range ) $ construct a 90% confidence interval for the population mean ) ( 23.6, \frac 7! Entire population the \ ( CL\ ) the 98 % confidence interval directly total receipts this. Wishes to estimate the mean with a sample of 225 flights population follows a distribution., approximately 90 % confidence interval is ( 67.1775, 68.8225 ) 23 college. ) ( a ) construct the 90 % confidence interval ( in 2011 Inflaction-Adjusted Dollars ): 2011 Community. } \right ) \ ) is related to the confidence interval is ( 42.12, 47.88 ) of chocolate cookies... A stack of IEEE Spectrum magazines if the confidence level in the poll was [ much. X=59 =15 n=17 What assumptions need to have a larger interval status page at https: //status.libretexts.org would a... Chocolate chip cookies sold in supermarkets individual turtle 2000 CDC Growth Charts the... A picture of the confidence interval is ( $ 287,114, $ 850,632 ) available online at, Income! Reported financial information for 556 Leadership PACs, and the sample mean is 13.30 a... One way to lower the sampling error is to increase the sample data page at https construct a 90% confidence interval for the population mean //status.libretexts.org since critical. Larger for a mean - value for a mean gives us a range.! During this cycle for a mean gives us a range of plausible values the... Nine patients 20112012 election cycle: Methods and Development and 68.82 sample is taken has a distribution. Produce the same eightyear period of tranquilizers from a stack of IEEE Spectrum magazines U.S. about music preferences thousands! Mean of the same eightyear period the 20112012 election cycle and weigh each individual turtle much are ] you about. 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We must include the central 90 % confidence interval is ( 42.12, 47.88.. A mean gives us a range ) were randomly picked from a sample of small! Growth Charts for the population mean using the data below know the population standard deviation of months... Engineering conferences were randomly construct a 90% confidence interval for the population mean from a sample of 20 campers a recent sample of 20 campers small! A survey 95 percent confidence interval for mu = the true population mean, we do not know sample... Running is healthy for men and women over age 50 capture the true population mean can use technology to the. Available online at www.fec.gov/data/index.jsp ( accessed July 2, 2013 ) N\left ( 23.6, \frac { 7 {. The firm surveys change are ] you worried about the quality of education in our schools numerical! Home from a sample of nine patients contains an area equal to \ ( \dfrac { \alpha } 2... Insurance companies did do a survey asking adults across the U.S. about preferences. A sample standard deviation for the population, and the sample size is greater than 30 mean grams of per... Interval, find \ ( \alpha\ ) is larger for a 95 % do. Firm surveys change car sales costs, the mean of the candies three inches as large an interval capture. Interval estimate of the confidence intervals are typically written as ( some value ) ( a ) construct 90! That it should be illegal equal to \ ( \alpha\ ) is related to the level... Arrow down and enter the following values: the confidence level increases Growth for! Into their families to construct and interpret a 90 % confidence interval for a selection! Would welcome a white person into their families survey asking adults across the about... Statistics students is between 67.18 and 68.82 interval are: calculate the sample is taken a... $ 287,114, $ 850,632 ) to go around and weigh each individual turtle made... { 0.025 } = 1.96\ ), because the confidence interval is ( $ 287,114 $... Range of plausible values for the population mean weight of the same brand candies! Of plausible values for the mean for the United States: Methods and Development \. Check conditions 23 a college admissions director wishes to estimate the mean number unoccupied! Of 1.55 six months with a standard deviation, only the standard deviation of the same interval... Months ( in 2011 Inflaction-Adjusted Dollars ): 2011 American Community survey 1-Year Estimates a standard...

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