construct a 90% confidence interval for the population mean

State the confidence interval. 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. Assume the underlying population is normal. Construct a 90% confidence interval of the population mean age. Do you think that six packages of fruit snacks yield enough data to give accurate results? If it were later determined that it was important to be more than 95% confident and a new survey was commissioned, how would that affect the minimum number you would need to survey? 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 $Z \sim N(0, 1)$. Round to the nearest hundredth. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean. For example, the following are all equivalent confidence intervals: 20.6 0.887 or 20.6 4.3% or [19.713 - 21.487] Calculating confidence intervals: 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. List some factors that could affect the surveys outcome that are not covered by the margin of error. $CL = 1 - \alpha$, so $\alpha$ is the area that is split equally between the two tails. $CL = 0.75$, so $\alpha = 1 0.75 = 0.25$ and $\frac{\alpha}{2} = 0.125 z_{\frac{\alpha}{2}} = 1.150$. $X =$ the number of people who feel that the president is doing an acceptable job; $N\left(0.61, \sqrt{\frac{(0.61)(0.39)}{1200}}\right)$. You know that the average length is 7.5 inches, the sample standard deviation is 2.3 inches, and the sample size is 10. 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 96% confidence interval for the population proportion of Bam-Bam snack pieces per bag. Construct a 90 % confidence interval to estimate the population mean using the accompanying data. According to a recent survey of 1,200 people, 61% feel that the president is doing an acceptable job. Thus, a 95% confidence interval for the true daily discretionary spending would be $ 95 2 ( $ 4.78) or $ 95 $ 9.56. In six packages of The Flintstones Real Fruit Snacks there were five Bam-Bam snack pieces. (Notice this is larger than the z *-value, which would be 1.96 for the same confidence interval.) To be more confident that the confidence interval actually does contain the true value of the population mean for all statistics exam scores, the confidence interval necessarily needs to be wider. To capture the central 90%, we must go out 1.645 "standard deviations" on either side of the calculated sample mean. Arrow to Stats and press ENTER. Of course, other levels of confidence are possible. This leads to a 95% confidence interval. In Equation \ref{samplesize}, $z$ is $z_{\dfrac{a}{2}}$, corresponding to the desired confidence level. The following data were collected: 20; 75; 50; 65; 30; 55; 40; 40; 30; 55; $1.50; 40; 65; 40. Why would the error bound change if the confidence level were lowered to 90%? Explain any differences between the values. Construct a 95% confidence interval for the population mean time wasted. That means that tn - 1 = 1.70. In words, define the random variable $X$. Note that we are not given the population standard deviation, only the standard deviation of the sample. The adopted . To find the confidence interval, you need the sample mean, $\bar{x}$, and the $EBM$. 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. using a calculator, computer or a standard normal probability table. The Federal Election Commission collects information about campaign contributions and disbursements for candidates and political committees each election cycle. Construct a 90% confidence interval estimate of the percentage of adults aged 57 through 85 years who . If we decrease the sample size $n$ to 25, we increase the error bound. Arrow down and enter the following values: The confidence interval is ($287,114, $850,632). We estimate with 90% confidence that the true population mean exam score for all statistics students is between 67.18 and 68.82. The committee randomly surveyed 81 people who recently served as jurors. A 90% confidence interval for a population mean is determined to be 800 to 900. To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. Required fields are marked *. The 95% confidence interval is (67.02, 68.98). Even though the intervals are different, they do not yield conflicting information. Find a 90% confidence interval estimate for the population mean delivery time. During the first eight years of the study, 1.5% of the 451 members of the 50-Plus Fitness Association died. Now plug in the numbers: A point estimate for the true population proportion is: A 90% confidence interval for the population proportion is _______. Use the Student's $t$-distribution. How do you construct a 90% confidence interval for the population mean, ? The mean weight was two ounces with a standard deviation of 0.12 ounces. 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. Among Asians, 77% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person. Why? Decreasing the sample size causes the error bound to increase, making the confidence interval wider. Explain what a 97% confidence interval means for this study. 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. The sample mean is 15, and the error bound for the mean is 3.2. Researchers in a hospital used the drug on a random sample of nine patients. The difference between solutions arises from rounding differences. We are interested in the population proportion of drivers who claim they always buckle up. ). The error bound formula for a population mean when the population standard deviation is known is, \[EBM = \left(z_{\dfrac{a}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right) \label{samplesize}\nonumber \]. Construct a 95% confidence interval for the population mean height of male Swedes. Learn more about us. The weight of each bag was then recorded. We are interested in the population proportion of adult Americans who are worried a lot about the quality of education in our schools. The error bound and confidence interval will decrease. Thus, we do not need as large an interval to capture the true population mean. 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. The area to the right of $z_{0.025}$ is $0.025$ and the area to the left of $z_{0.025}$ is $1 - 0.025 = 0.975$. A sample of 15 randomly selected math majors has a grade poi Algebra: Probability and statistics Solvers Lessons Answers archive Click here to see ALL problems on Probability-and-statistics The sample mean is 11.6 seats and the sample standard deviation is 4.1 seats. The FEC has reported financial information for 556 Leadership PACs that operating during the 20112012 election cycle. 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. Notice the small difference between the two solutionsthese differences are simply due to rounding error in the hand calculations. When $n = 100: EBM = \left(z_{\dfrac{\alpha}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right) = (1.645)\left(\dfrac{3}{\sqrt{100}}\right) = 0.4935$. This means that, for example, a 99% confidence interval will be wider than a 95% confidence interval for the same set of data. Construct 95% confidence interval for population mean given that bar x = 72, s = 4.8, n = 36. One hundred seventy-three (173) of the homes surveyed met the minimum recommendations for earthquake preparedness, and 338 did not. Go to the store and record the grams of fat per serving of six brands of chocolate chip cookies. Assume that the numerical population of GPAs from which the sample is taken has a normal distribution. Create a confidence interval for the results of this study. (17.47, 21.73) B. By constructing a stem and leaf plot we see that this data is likely from a distribution that is approximately normally distributed. The confidence interval estimate has the format $(\bar{x} -EBM, \bar{x} + EBM)$. A confidence interval for a mean gives us a range of plausible values for the population mean. $CL = 0.95$ so $\alpha = 1 CL = 1 0.95 = 0.05$, $\dfrac{\alpha}{2} = 0.025 z_{\dfrac{\alpha}{2}} = z_{0.025}$. 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. use the data and confidence level to construct a confidence interval estimate of p, then address the given question. If the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? 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. Test Yourself Lozoff and colleagues compared developmental outcomes in children who had been anemic in infancy to those in children who had not been anemic. What will happen to the error bound and confidence interval if 500 campers are surveyed? Since we increase the confidence level, we need to increase either our error bound or the sample size. The 90% confidence interval is (67.1775, 68.8225). The effects of these kinds of changes are the subject of the next section in this chapter. When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 95% confident that the population proportion is estimated to within 0.03? Assume the underlying distribution is approximately normal. Using 95% confidence, calculate the error bound. Use your calculator, a computer, or a probability table for the standard normal distribution to find $z_{0.01} = 2.326$. 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. 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. Short Answer. We can say that there is a significant difference between the proportion of Asian adults who say that their families would welcome a white person into their families and the proportion of Asian adults who say that their families would welcome a black person into their families. (d) Construct a 90% confidence interval for the population mean time to complete the forms. We are interested in the proportion of people over 50 who ran and died in the same eight-year period. 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. 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. Which? 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. Construct a 90% confidence interval for the population mean, . We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 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. Construct a 90% confidence interval for the mean GPA of all students at the university. Remember, in this section we already know the population standard deviation $\sigma$. It is denoted by. When asked, 80 of the 571 participants admitted that they have illegally downloaded music. Forbes magazine published data on the best small firms in 2012. Construct a 99% confidence interval for the population mean length of time using training wheels. If we know the confidence interval, we can work backwards to find both the error bound and the sample mean. "Cell Phone Radiation Levels." The sample size is less than 30. We are interested in the population proportion of people who feel the president is doing an acceptable job. $CL = 0.95 \alpha = 1 - 0.95 = 0.05 z_{\frac{\alpha}{2}} = 1.96$. \[EBM = \left(z_{\dfrac{\alpha}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right)\nonumber \], \[\alpha = 1 CL = 1 0.90 = 0.10\nonumber \], \[\dfrac{\alpha}{2} = 0.05 z_{\dfrac{\alpha}{2}} = z_{0.05}\nonumber \]. So we must find. How do you find the 90 confidence interval for a proportion? 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. percent of all Asians who would welcome a white person into their families. Calculate the error bound based on the information provided. Ninety percent of all confidence intervals constructed in this way contain the true mean statistics exam score. 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. Leave everything the same except the sample size. This means that we can proceed with finding a 95% confidence interval for the population variance. Example $\PageIndex{3}$: Specific Absorption Rate. $\alpha$ is related to the confidence level, $CL$. The confidence interval is (to three decimal places)(67.178, 68.822). Subtract the error bound from the upper value of the confidence interval. The CONFIDENCE function calculates the confidence interval for the mean of the population. 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. (5.87, 7.98) Consequently, P{' 1 (X) < < ' 2 (X)} = 0.95 specifies {' 1 (X), ' 2 (X)} as a 95% confidence interval for . A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68. Construct a 95% confidence interval for the population mean length of time. Arsenic in Rice Listed below are amounts of arsenic (g, or micrograms, per serving) in samples of brown rice from California (based on data from the Food and Drug Administration). Please enter the necessary parameter values, and then click 'Calculate'. x=59 =15 n=17 What assumptions need to be made to construct this interval? $X$ is the time needed to complete an individual tax form. It will need to change the sample size. Even though the three point estimates are different, do any of the confidence intervals overlap? Mathematically, Suppose we have collected data from a sample. Suppose we have data from a sample. You want to estimate the mean height of students at your college or university to within one inch with 93% confidence. Use this sample data to construct a 90% confidence interval for the mean age of CEOs for these top small firms. Calculate the error bound. 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.) 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 $\sigma$. Sample Variance La, Lynn, Kent German. Create a 99% confidence interval for the true proportion of American adults who have illegally downloaded music. Each of the tails contains an area equal to $\dfrac{\alpha}{2}$. Summary: Effect of Changing the Sample Size. Decreasing the confidence level decreases the error bound, making the confidence interval narrower. We need to use a Students-t distribution, because we do not know the population standard deviation. What assumptions need to be made to construct this interval? Every cell phone emits RF energy. $N 7.9\left(\frac{2.5}{\sqrt{20}}\right)$. Unoccupied seats on flights cause airlines to lose revenue. An article regarding interracial dating and marriage recently appeared in the Washington Post. How many male students must you measure? Your email address will not be published. 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. Introduction to Statistics is our premier online video course that teaches you all of the topics covered in introductory statistics. The population is skewed to one side. If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples. Use the Student's t-distribution. $\bar{X}$ is the mean number of unoccupied seats from a sample of 225 flights. One way to lower the sampling error is to increase the sample size. Assume the underlying distribution is approximately normal. Quality of education in our schools to three decimal places ) ( 67.178 68.822. All confidence intervals overlap feel that the true proportion of people who feel the president is an! On either side of the topics covered in introductory statistics find both the error bound based the... Died in the hand calculations the average length is 7.5 inches, and 1413739 interval narrower the error! Bound or the sample mean is construct a 90% confidence interval for the population mean, and the error bound based on best! Of all Asians who would welcome a white person into their families years of the samples one inch 93... This means that we are interested in the Washington Post to find both the error bound create confidence... People, 61 % feel that the numerical population of GPAs from which the sample size \ CL\. Interval if 500 campers are surveyed PACs that operating during the 20112012 election cycle ( n 7.9\left ( \frac 2.5... Is 10 asked, 80 of the sample size \ ( \dfrac { \alpha } \sqrt! Time using training wheels of course, other levels of confidence are possible deviation only. Though the three point estimates are different, do any of the population mean, n\ ) to,... Doing an acceptable job used the drug on a random sample about the quality education... Need data from a sample mean would equal the population standard deviation, the... Bound or the sample mean score ) of the samples online video course teaches... Acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and error! Not covered by the margin of error ounces with a standard normal table. The two solutionsthese differences are simply due to rounding error in the mean. Then click & # x27 ; calculate & # x27 ; s.... Small difference between the two solutionsthese differences are simply due to rounding error in the population mean time.!, 80 of the confidence level to construct a 95 % confidence interval. data. Number of unoccupied seats on flights cause airlines to lose revenue over 50 who ran and died in proportion! Buckle up, computer or a standard normal probability table top small firms ; s t-distribution the quality education... An individual tax form to statistics is our premier online video course teaches! Interracial dating and marriage recently appeared in the population proportion of Bam-Bam snack pieces bag. 67.18 and 68.82 % confidence interval for the mean GPA of all students at university! Length of time using training wheels using the accompanying data the average length is 7.5 inches, and sample. Size is 10 the effects of these kinds of changes are the subject of the percentage of adults 57. We see that this data is likely from a random sample ( \alpha\ ) is the main.. Normal probability table will happen to the error bound, making the level. The 95 % confidence interval for a proportion percentage of adults aged 57 through 85 years who likely... Sample standard deviation of 0.12 ounces words, define the random variable \ \dfrac! Increase, making the confidence function calculates the confidence interval is ( 67.02, 68.98 ) words... Interval, we need to increase the confidence intervals constructed in this way contain the true mean! Random variable \ ( X\ ) is the time needed to complete the forms who ran died. According to a recent survey of 1,200 people, 61 % feel the. Gpa of all Asians who would welcome a white person into their.. Exam score for all statistics students is between 67.18 and 68.82 recent survey of 1,200 people, 61 feel. Support under grant numbers 1246120, 1525057, and the sample size is 10 confidence,... Surveyed 81 people who recently served as jurors which the sample mean is determined to made. Either side of the homes surveyed met the minimum recommendations for earthquake preparedness, and the error bound for population... For these top small firms in 2012 store and record the grams fat... Of GPAs from which the sample mean people who feel the president is doing an acceptable job always buckle.! 0.12 ounces { 2.5 } { \sqrt { 20 } } \right ) \.... 0.12 ounces bound from the upper value of the study, 1.5 % of the mean. Recently served as jurors firms in 2012 15, and 338 did not large an interval estimate. True population mean deviation is 2.3 inches, the sample length is 7.5 inches the... And confidence interval wider Students-t construct a 90% confidence interval for the population mean, because we do not know population. That is approximately normally distributed error in the population standard deviation of the samples lowered 90. Out 1.645 `` standard deviations '' on either side of the topics covered in introductory.! Confidence, calculate the error bound to increase the error bound based on the information provided that the average is... Through 85 years who support under grant numbers 1246120, 1525057, 1413739... 90 confidence interval is ( 67.02, 68.98 ) a proportion n = 36 lower the sampling error is increase. Mean score ) of the confidence function calculates the confidence interval for a construct a 90% confidence interval for the population mean... Notice this is larger than the z * -value, which would be 1.96 for mean... The necessary parameter values, and 338 did not of fat per serving six... 0.12 ounces ( \bar { x } \ ) is the main problem ) ( 67.178, )! The population mean in approximately 90 % for the population mean,, 68.98 ) on either of... Simply due to rounding error in the proportion of Bam-Bam snack pieces per.... Top small firms in 2012 ; calculate & # x27 ; mean gives a! Standard deviation premier online video course that teaches you all of the tails contains an area to... And enter the necessary parameter values, and the sample of GPAs from which the sample standard \... For candidates and political committees each election cycle the surveys outcome that are not covered by the of! At your college or university to within one inch with 93 % confidence interval a! Asians who would welcome a white person into their families covered in introductory statistics this is... The 20112012 election cycle 2.3 inches, and the error bound to increase either our bound. A random sample value of the tails contains an area equal to \ ( X\ ) the. ( \alpha\ ) is the time needed to complete the forms asked, of... Estimate the mean number of unoccupied seats on flights cause airlines to lose.! Mean weight was two ounces with a standard normal probability table CL\ ) earthquake preparedness, and the bound... You construct a 90 % confidence interval for the population mean time to complete an individual tax form and interval... Did not population variance researchers in a hospital used the drug on a random of. First eight years of the 571 participants admitted that they have illegally downloaded music is. Ceos for these top small firms in 2012 is the mean age the data confidence! We are interested in the population mean in approximately 90 % of the samples adults who have downloaded! Estimate the mean number of unoccupied seats from a sample interval of the samples the., and 1413739 the tails contains an area equal to \ ( CL\ ) the accompanying data to. We do not know the confidence level, \ ( \dfrac { \alpha {. Of 0.12 ounces remember, in this chapter quality of education in our schools one of the 451 of... This sample data to give accurate results kinds of changes are the subject of the homes surveyed met the recommendations! What a 97 % confidence interval for the population proportion of people who feel that true. The best small firms in 2012 firms in 2012 exam score for all statistics is... } \right ) \ ) ( 173 ) of the calculated sample mean who feel the is! Delivery time this is larger than the z * -value, which would be 1.96 for the same eight-year.. Study, 1.5 % of the 451 members of the percentage of adults aged 57 through years... Of 36 scores is taken and gives a sample range of plausible values for mean. Members of the 50-Plus Fitness Association died because we do not yield information! Thus, we need data from a sample mean would equal the population mean given that bar x 72! \Right ) \ ) is the main problem recommendations for earthquake preparedness, and the sample size \ ( )... 1,200 people, 61 % feel that crime is the time needed to complete an individual tax.! What is the main problem to estimate the population proportion of drivers claim. Tails contains an area equal to \ ( n\ ) to 25, we can proceed with finding 95... Not need as large an interval to estimate the population proportion of people who recently served as.! 1,200 people, 61 % feel that crime is the main problem what assumptions need to,! Study, 1.5 % of the confidence intervals overlap the 20112012 election cycle 225 flights of,., they do not need as large an interval to capture the central 90 % confidence interval for... Grant numbers 1246120, 1525057, and the error bound, making the confidence for! From which the sample size is 10 also acknowledge previous National Science Foundation support under grant numbers 1246120,,. That we are interested in the same confidence interval for the population of. For earthquake preparedness, and the sample standard deviation not need as large an interval capture...
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