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# ProbStats MATH 1530

ETSU

GPA 3.5

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This 3 page Class Notes was uploaded by Ms. Ismael Spinka on Sunday October 11, 2015. The Class Notes belongs to MATH 1530 at East Tennessee State University taught by Deborah Hosler in Fall. Since its upload, it has received 32 views. For similar materials see /class/221400/math-1530-east-tennessee-state-university in Mathematics (M) at East Tennessee State University.

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Date Created: 10/11/15

MATH153004071517 Probability amp Statistics Fall 2004 Week 12 Using Con dence Intervals to Estimate Unknown Population Para meters Estimation of an unknown population proportion p The best choice for an estimator ofan unknown population proportion is the sample proportion phat m The Associated Press Dec 16 1991 reported that in a sample of 507 adult US citizens only 142 correctly described the Bill of Rights as the rst ten amendments to the US Constitution WHAT is the variable ofinterest It is whether or nota citizen correctly describes the Bill of Rights a categorical variable It doesn t make sense to calculate a mean in this case butwe can calculate a proportion Remember that there are certain assumptions that should be checked before we carry out any inference concerning proportions Ifwe can meet these assumptions then the amount of con dence with which we are constructing the intenals should be fairly accurate What are the 3 conditions that need to be met see p 475 in BPS A First calculate the sample proportion phat and use it to nd a 95 confidence interval for the true proportion of all US adults that could give a correct description of the Bill of Rights The formula you need is A t A 1 A p i Z p p n B The margin of error calculated in part A is about 4 What size sample would we need to produce a 95 Cl with a smaller margin of error of m 0025 Assume that we can use the result ofthe pilot studyquot described above to give us a hint of the value of the proportion of successes in the population that is plug in the phat calculated above for p in the formula given below NOTE In this case we have speci c instructions about the value of p to use but in a situation where we do not have a clue about the real population proportion we should use 05 for p Here is the formula to use n2plp Estimation ofan unknown population mean mu The most natural and intuitive choice for the estimator of mu is the sample mean xbar but it is also nice that the mathematics ensures that xbar is unbiased and has low variability as an estimator of mu take my word for it and I ll spare you the mathematics However before we start estimating we must examine the distribution of variable X in the population as well as the sampling distribution of x bar mile One of the everyday uses of estimation is to assure quality control in manufacturing and industrial settings in which samples are collected as often as every day or every shift in order to check that a certain process is running according to specification We are given n 16 measurements of a critical dimension in mm on a sample needs to be an SRS of auto engine crankshafts WHAT is the variable of interest It is some dimension of a crankshaft which is measured in millimeters a quantitative variable The process mean is supposed to be mu 224000 mm but can drift away from this target during production We have reason to believe that individual measurements in the population are distributed Normally but the rst thing we should do is some exploratory data analysis for the sample Let s look at a histogram and the basic statistics Sixteen Crankshaft Measurements Frequency n w I I I I I I I I 22390 22393 22396 223 99 22402 22405 224 08 22411 C1 Minitab Output Descrip Van39able N Mean Median TrMean StDev SE Mean Minimum Maximum Q1 Q3 dimension mm 16 224002 223988 22400 00618 00155 223902 224120 22396 22405 Notice that the distribution of measurements in the sample is roughly symmetric and there do not appear to be any outliers There is nothing here that contradicts our assumption of normality We should proceed with the estimation Also note that the estimatoris xbar but the speci c estimate is the speci c value of the sample mean 224002 mm A We wish to give a 90 confidence interval for the process mean at the time these crankshafts were produced Suppose for now that we know the standard deviation for the population of all crankshafts produced is sigma 0060 The formula we need to carry out this estimation is B However we cannot assume that the population standard deviation will always be known There is another estimation technique that uses the tStudent distribution instead ofthe Normal that can help us overcome this obstacle With the tStudent model and the sample standard where t is determined not only by the level of con dence but also by n 1 degrees of freedom just take the sample size and subtract one to find the proper row in the ttable Notice that this formula calls for the sample mean xbar which is given in the Minitab output It also calls for the sample standard deviation s and the sample size n Plug those values into the formula and recalculate the 90 con dence intenal The critical value oft can be found in the ttable in the column located above the desired con dence level and in the rowfor n 1 deviation s the Cl formula looks like this degrees of freedom C Also calculate a 95 confidence interval and a 99 confidence interval for mu using the formula based on the t distn39bution Are these intervals narrower or wider than the one previously calculated in part B 95 CI 99CI D Now calculate a 90 confidence interval for mu using the sample mean given above but assuming that the sample size is n 36 measurements Is the intenal narrower orvvider than the intenal constructed in part B This is an example of some Minitab Output for the confidence intervals you calculated in parts B and C above 90 T Confidence Interval for process mean of crankshaft measurements Variable N Mean StDev SE Mean 900 CI dimension 16 224002 0062 0015 223975 224029 95 T Confidence Interval for process mean of crankshaft measurements Variable N Mean StDev SE Mean 950 CI dimension 16 224002 0062 0015 223969 224035 99 T Confidence Interval for process mean of crankshaft measurements Variable N Mean StDev SE Mean 990 CI dimension 16 224002 0062 0015 223956 224047 For you information Minitab Instructions Enter the data on Crankshaft Dimensions into one column ofa minitab worksheet Go to Stat gt Basic Statistics gt 1Sample t Variables select the column with the Crankshaft Dimensions Click on 09tions button Confidence level it is 950 by default but you may specify otherlevels Click OK Click OK

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