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# STAT ANA FOR BUS I MSIT 3000

UGA

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This 11 page Class Notes was uploaded by Hallie Heaney III on Saturday September 12, 2015. The Class Notes belongs to MSIT 3000 at University of Georgia taught by Werner in Fall. Since its upload, it has received 31 views. For similar materials see /class/202140/msit-3000-university-of-georgia in Management Sciences And Information Technology at University of Georgia.

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Date Created: 09/12/15

Business Statistics A First Course Chapter 9 Con dence Intervals for Proportions 92 The Sampling Distribution for Proportions Because each sample is different he observed proportion of successes f9 varies from sample to sample However the 13 s vary in a way that we can model and predict The distribution of many 15 s from a large number of samples is called the sampling distribution of f9 The sampling distribution of f9 is a Normal distribution centered at the true proportion p The standard deviation of the sampling distribution also called the standard error SESD 2011 Pearson Education Inc 39 quot 39 95 A Confidence Interval Example A Gallup Poll found that 153 out of 1023 people thought economic conditions were getting better The sampling distribution of 3 follows a Normal model with center at the truep and standard error 0011 51513 NOISE 2115 0011 l I l l l l i p msa 121 0022 p 0011 p p0011 p0022 p0033 2011 Pearson Education Inc 95 A Confidence Interval Because the distribution is Normal 095 of similar samples produce 13 within 196 SEs of the truep 0025 outside either side We are 95 sure 13 is within 196 x 0011 ofp 4 3 2 1 U 1 2 3 4 The formula for the confidence interval is 13i2SE SE Wo011 39 n 1023 015 i 196 0011 For 95 confidence 0128 0172 2011 Pearson Education Inc 39 39 39 39 39 quot 95 A Confidence Interval What Does 95 Confidence Really Mean Below are confidence intervals produced by simulating 20 samples The purple dots are the 13 of the 20 samples the orange lines show the confidence intervals The green line is the true p Not all intervals capture the true proportion 5 of the time the confidence interval misses the true p P ropor on 2011 Pearson Education Inc 96 Margin of Error Certainty vs Precision The confidence interval is given by The width of the interval on each side of f is called the margin of error ME The margin of error is half the width of the confidence interval The wider the interval the larger the margin of error ME gives the maximum error between p and 15 for the given confidence level In general the confidence interval can be written where ME 2 SE Note that for a given confidence interval the Margin of Error is alay If he width For example 04 06 has ME 01 2011 Pearson Education Inc 96 Margin of Error Certainty vs Precision The more confident we want to be about including the truep the larger the margin of error We can be very confidentp is inside a very wide interval but we can t be very confidentp is inside a narrow interval Every confidence interval is a balance between confidence and interval width precision The greater the confidence the wider the interval must be to capture the true p A narrow interval will have a low confidence of capturing the true p Wide confidence intervals have a high chance of capturing p but also a large margin of error Narrow intervals have a small margin of error but a smaller chance of capturing p 2011 Pearson Education Inc 97 Interpretation and Critical Values How do you interpret a confidence interval We are 95 confident that between 128 and 172 of US adults thought the economy was improving To change the confidence we must change 2 This will change the width of the interval the chance of capturing p Some common con dence levels and their associated critith mines C I 90 1645 95 1960 99 2576 2011 Pearson Education Inc 98 Conditions Q Can we always use this procedure to construct a confidence interval i No there are 4 conditions 1 Independence The samples must be independent of each other There should not be any common factors such as the same family groups of friends snowstorm affecting everybody in the same region etc 2 Randomization The samples must be chosen randomly This helps to avoid bias both conscious and unconscious 2011 Pearson Education Inc 98 Assumptions and Conditions 3 10 Condition The population must be at least 10 times larger than the sample 4 Sample size The number of successes np 2 10 and also the number of failures mg 210 r Effect of Sample Size Because H is in the denomi na tor of SD the larger the sample the smaller the standard deviatioiL V sTe need a small standard deviation to make sound business decisions but larger samples cost more That tension is fundamen tal issue in statistics 2011 Pearson Education Inc 99 Sample Size To get a narrower interval for the same confidence we must choose a larger sample Example Suppose a company wants to offer a new service and wants to estimate the proportion of customers to within 3 who are likely to purchase this new service with 95 confidence How large a sample do they need 003 is the margin of error and for 95 confidence 2 196 A 2 196 2 pq Z AA 39 AA MEZ gtn 72 ME m 003 W What is the value of f9 and c 2011 Pearson Education Inc

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