Confidence Intervals QMB Exam 2 Study Guide Point Estimates Single value that best describes population of interest o Sample mean is a point estimate of unknown population mean o Easy to calculate but do not provide any information on their accuracy Interval estimate provides additional information about variability o Confidence Interval range of where interval can be placed between For the confidence interval must know where the standard deviation is coming from sample of the population Conditions for confidence intervals o Sample size at least 30 or population follows normal distribution o The population standard deviation is known Standard error of the mean Population std deviation the square root of the sample size n UCLx upper limit X Za 2Ox LCLx lower limit X Za 2Ox O Standard deviation Za 2 critical z score is dictated by the level of confidence Critical z score o a is known as significance level Example if a 0 1 then Za 2 Z0 05 1 645 value encloses 90 of the area under the normal distribution between which leaven 5 in each tail 1 2 05 on each side of the graph tails o Example Given n 35 X 145 5 O 30 2 compute 90 confidence interval Standard error of the mean Population std deviation the square root of sample size n 30 2 35 5 105 Za 2 for 90 Z0 05 1 645 on z table UCLx upper limit X Za 2Ox o UCLx 145 5 1 645 5 105 153 90 LCLx lower limit X Za 2Ox o LCLx 145 5 1 645 5 105 137 10 Based on the sample mean of 145 5 we are 90 confident that the population mean is between 137 10 and 153 90 Margin of Error MEx is the amount added and subtracted to the point estimate to form the confidence interval o MEx Za 2Ox Increasing sample size while keeping the confidence interval level constant will reduce the margin or error resulting in a narrower more precise confidence interval Significance level probability that any given confidence interval will not contain the true population o Confidence level width increases when confidence level increase Confidence Level Significance level 80 90 95 98 99 20 10 5 2 1 Z score 1 28 1 645 1 96 2 33 2 575 Calculating confidence intervals when population standard deviation is unknown When standard deviation of population is unknown we substitute the sample standard deviation s in place to calculate the standard error o Formula for approximate standard error s n o T distribution issued in place of normal probability distribution when sample standard deviation is used Bell shaped Mean 0 Shape of curve depends on degrees of freedom df n 1 The critical score for t distribution is greater than the critical z score for sample confidence level o AS number of degrees of freedom increases shape becomes similar to normal distribution Formula for confidence intervals remain the same except we use the standard error of the sample mean UCLx upper limit X ta 2Oxbar LCLx lower limit X ta 2Oxbar o Sample mean 2 65 per gallon n 15 xbar 2 64 s 0 087 df 15 Example 1 14 Approximation standard error of the mean Approximate standard error s n o 087 15 0 225 90 confidence interval confidence level 0 90 Significance level 1 0 9 1 ta 2 t0 05 1 761 use the t table distribution o UCLx upper limit X ta 2Oxbar o LCLx lower limit X ta 2Oxbar LCL 2 64 1 761 0 0225 2 60 UCL 2 64 1 761 0 0225 2 68 Since the claim that the population mean is 2 65 is supported since the interval 2 6 2 68 includes the value 2 65 Since population shape is not specified and sample size n is less than 30 we must assume the population Calculating confidence intervals for proportions Proportions data follows the binomial distribution which can be approximated by normal distribution under the following conditions o n 5 and n 1 5 o Where probability of success in the population n sample size o An interval estimate Formula for sample proportion P x n o X number of observations of interest success o N sample size Standard deviation of proportion 1 n Where population proportion When is unknown o p 1 p n p sample proportion n sample size Confidence Interval o UCLp upper limit X Za 2Op o LCLp lower limit X Za 2Op o Margin of error Za 2Op Example o 62 of individuals n 225 163 x Construct a 95 confidence interval for 2010 P x n 163 225 7244 SD p 1 p n o 7244 1 7244 225 0298 o 95 confidence interval confidence level 0 95 o Significance level 1 0 95 0 05 o Za 2 Z0 025 1 96 use z table o UCLp upper limit X Za 2Op o LCLp lower limit X Za 2Op LCL 7244 1 96 0 0298 6660 UCL 7244 1 96 0 0298 7828 Determining Sample size o Increase of sample size all else constand reduces the margin of error and provides a narrower confidence interval Sample size needed to achieve a specific margin of error can be calculated given Confidence interval level Population standard deviation o Solving for n formula for sample size neede to estimate population mean Smallest sample size needed always round up o If population standard deviation is unknown use sample mean distribution For population proportion sample size needed formula Will need to use t distribution instead Calculating Sample Size In order to calculate the required sample size to estimate the population proportion we need to know p sample population o Select a pilot sample and use sample proportion p If its no possible to select a pilot sample set p 0 50 Provides most conservative estimate for sample size o Guarantee largest sample size Example o Sample size needed to construct a 95 confidence interval for the population mean ME 10 0 80 Significance level 1 0 95 0 05 Z0 05 1 96 N Za 2 202 MEx o N 1 96 2 80 2 10 2 245 86 always round up o N 246 ME 15 0 80 Z 1 96 N Za 2 202 MEx o N 1 96 2 80 2 15 2 109 3 always round up o N 110 Example o Sample size needed to construct 95 of population portion MEp 0 04 p 0 3 Significance level 0 05 z 1 96 N Za 2 2p 1 p ME 2 o N 1 96 2 0 3 0 7 0 04 2 504 21 always round up o N 505 Calculating Confidence Intervals for finite population o Finite population overestimates the standard error when sampling without replacement Adjust standard error by using finitite population correction Correction factor N n N 1 factor o Add to UCL and LCL o If you were given s instead of standard deviation use t distribution instead of z distribution Assume dealing with normal distribution underlying population Degree of freedom n 1 when using t distribution Std dev n N n …