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# Quant Meth Agribus AEB 5516

UF

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This 5 page Class Notes was uploaded by Weldon Rau I on Friday September 18, 2015. The Class Notes belongs to AEB 5516 at University of Florida taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/206593/aeb-5516-university-of-florida in Agricultural Economics And Business at University of Florida.

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Date Created: 09/18/15

SECTION 5 CONFIDENCE INTERVALS Con dence Intervals 039 known A con dence interval is a range of values constructed about a point estimate of the mean or proportion such the probability that the unknown or hypothesized population mean proportion lies somewhere in the interval is known For example a 95 con dence interval for the mean may be denoted as PL1 S u S U51 095 The key to constructing con dence intervals thus involves determining the values for LEI and U51 In general the level of con dence de ning the interval is symbolized by l a X 100 Where a is the proportion of the probability lying outside the con dence interval in the tails of the distribution Due to the symmetry of the normal distribution a 2 of the proportion lies on each upper and lower tail The con dence interval for the mean is given by J r 2m or X Zm ssthzaZ Where Z is there critical value of the standard normal distribution containing a 2 aZ probability in each tail For a 95 percent con dence interval a 05 a 2 0025 and the critical value of Zl96 Thus a 95 con dence interval for the population mean is given by X i96 sys196i J S Example The inspecting division of the Lee County Weights and Measures Department wants to estimate the actual amount of so drink in 2liter bottles at the local bottling plant of a nationally known soft drink company The bottling plant informed the inspection division that the population standard deviation for 2liter bottles is 005 liters A random sample of 100 2liter bottles had a sample mean of 199 liters a Construct a 95 con dence interval for the population mean for the amount of soft drink in each bottle b Must you assume the population of soft drink fill is normally distributed Explain c Explain why a value of 202 liters for a single bottle is not unusual even though it is outside the calculated con dence interval d Suppose the sample mean is 197 liters What is your answer to a Answer Con dence intervals 039 unknown Estimating the variance Recall that the population variance is a measure of dispersion defined as 0392 VarX EX u2 An intuitive estimator for this variance based on a random sample of n observation is X 22 n quot2 039 which is simply the average of the squared deviations of the sample about the sample mean While this estimator makes sense intuitively it provides a biased estimate of the population variance Thus E62 2 0392 The reason for this is related to the concept of degrees of freedom In any random sample each observation in essence carries one piece bit of information about the population Thus a sample of n contains n bits of information or 71 degrees of freedom Each time the sample is used to estimate some population parameter e g u one degree of freedom is lost Thus when a sample of n observations is used to estimate the sample mean one degree of freedom is lost As the mean is used in estimating the population variance based on the sample there are only rt J degrees of freedom available Thus the unbiased estimate of the population variance based on a sample of n observations is given by 2X J S2 11 n l And the estimated standard deviation is given by The Student s t Distribution Constructing confidence intervals when the population variance is unknown is analogous to the situation when the variance is known with one exception When the unknown estimate of the variance standard deviation is replace by the estimate S2 S the expression is no longer distributed as a standard normal random variable Instead it is distributed as a Student s t random variable with rt J degrees of freedom The tdistribution has the same shape as the normal distribution but with fatter tails This is essence serves to widen confidence intervals to re ect the fact that an estimate of the population standard deviation is being used rather than the population standard deviation As the sample size grows the tdistribution converges to the standard normal distribution A l a percent confidence interval is thus given by Or Where tH is the critical value of the tdistribution with nl degree of freedom for an area of a 2 in the upper tail of the tdistribution Example One operation of a mill is to cut pieces of steel into parts that are used later in the frame for front seats in an automobile The steel is cut with a diamond saw and require the parts to be within 0005 inch of the length specified by the automobile company The measurements reported contained in the section 5 Excel file is the difference in inches between the actual length of the steel part as measured by a laser measurement deVice and the specified length pf the part eg 0002 represents a steel part that is 0002 shorter than the specified length a Construct a 95 confidence interval estimate of the mean difference between the actual and specified length the steel part b What assumptions must be made about the population distribution in a Answer Con dence Intervals for the Proportion Con dence intervals for proportions are given by Pig n P Zm l snstaZ M n 71 Example 01 A large number of companies are trying to reduce the cost of prescription drug bene ts by requiring employees to purchase drugs through a mandatory mailorder program In a survey of 600 employers 126 indicated they either had a mandatory mailorder program in place or were adopting one by the end of 2008 a Construct a 95 percent con dence interval for the population proportion of employers who had or were contemplating adopting a milorder program b Construct a 99 con dence interval Answer

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