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# Controlling Confidence Interval Length

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Example 9 12 Sample Size Selection for Estimating Reaction to New Sandwich Controlling Confidence Interval Length Objective To find the sample size of customers required to achieve a sufficiently narrow confidence for the mean rating of the new sandwich 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 9 10 9 11 9 13 9 14 9 15 Background Information The fast food manager in Example 9 2 surveyed 40 customers each of whom rated a new sandwich on a scale of 1 to 10 Based on the data a 95 confidence interval for the mean rating of all potential customers extended from 5 739 to 6 761 for a half length of 6 761 5 379 2 0 511 How large a sample would be needed to reduce this half length to approximately 0 3 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 9 10 9 11 9 13 9 14 9 15 Confidence Intervals Confidence intervals are a function of three things the data in the sample We have control over the data by using the various random sampling plans to reduce variability An area of statistics called experimental design suggests how to perform experiments to obtain the most information from a given amount of sample data the confidence level This effect is clear as the confidence level increases the length of the confidence interval increases as well the sample size s The most obvious way to control confidence interval length is to choose the size of the sample appropriately 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 9 10 9 11 9 13 9 14 9 15 Sample Size Sample size selection must be done before a sample is observed Sample size estimation formula 1 96 1 597 n 0 3 2 z multiple x est n B 2 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 9 10 9 11 9 13 9 14 9 15 Calculations The formula for n uses three inputs the z multiple which is 1 96 for a 95 confidence level the prescribed confidence interval half length B which is 0 3 and an estimate sigmaest of the standard deviation The final input must be guessed but for this example with a sample size 40 we can use the observed sample standard deviation of 1 597 which can be determined with the STDEV function 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 9 10 9 11 9 13 9 14 9 15 Calculations continued The formula yields a rounded result of n 109 The claim then is that the manager surveys 109 customers with a 95 confidence interval will have the approximate half length of 0 3 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 9 10 9 11 9 13 9 14 9 15 Calculations continued The same calculations can be done using the Sample Size Selection procedure of Excel s StatPro add in Just select the menu item and select the parameter to analyze and enter the requested values A message telling you the required sample size is displayed 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 9 10 9 11 9 13 9 14 9 15 Question What if the manager was at the planning stage and didn t have a preliminary sample of size 40 What standard deviation estimate should she use for est The manager basically has three choices she can base her estimate of the standard deviation on historical data assuming that relevant historical data are available she can take a small preliminary sample of size 20 say just to get an estimate of the standard deviation she can simply guess a value for the standard deviation not recommended but there are some cases where it is the only feasible option 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 9 10 9 11 9 13 9 14 9 15

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