LAST TIME POINT ESTIMATION we know how to check if a sample mean X is statistically significantly different from a hypothesized population mean single value that represents the best estimate of a population value Introduction to Statistics in Psychology PSY 201 Professor Greg Francis Lecture 21 but sometimes we have no idea what is Estimation of population mean The great census debate when we want to estimate the best point estimate is the sample mean X but the estimate depends on which sample we select we would like to be able to estimate using the sample data we have Frequency 0 03 statistical estimation 0 02 0 01 0 0 20 40 60 80 100 120 Sample Mean 2 3 INTERVAL ESTIMATION KNOWN INTERVAL ESTIMATION we get a better idea of the value of by considering a range of values that are likely to contain to demonstrate our technique suppose we have a population of scores with 43 10 we will show how to build up confidence intervals using the properties of the sampling distribution of the mean from the population we get the sampling distribution for samples of size n 400 with X 0 50 n with the sampling distribution we can calculate using the standard normal table that 95 of all sample means will lie between 42 02 and 43 98 0 8 0 6 but since we do not really know the value of we must estimate it 0 8 0 6 0 8 0 4 0 4 0 6 0 2 0 2 0 4 0 0 41 42 43 44 45 0 2 41 Sample Mean 42 43 Sample Mean 0 41 42 43 44 45 Sample Mean 4 5 6 44 45 CONFIDENCE INTERVALS LEVEL OF CONFIDENCE CONFIDENCE INTERVAL construct an interval around the observed statistic X degree of confidence that computed interval contains suppose we calculate X 44 6 usually complement of level of significance CI X tcv sX CI statistic critical value standard error of the statistic level of confidence is 1 CI X tcv sX where calculating the critical value tcv is the same X is the sample mean tcv is the critical value using the appropriate t distribution for the desired level of confidence sX is the estimated standard error of the mean s sX n the confidence interval is then CI95 44 6 1 9659 0 50 CI95 43 61705 45 58295 e g for 0 05 1 0 95 and tcv 1 9659 using the inverse t distribution calculator with df n 1 399 7 8 9 CONFIDENCE INTERVAL EXAMPLE CONFIDENCE INTERVAL this means we are 95 confident that the interval 43 62 45 58 contains the unknown value Guess the height of this room in feet and write down your guess on the papers going around the room CI X tcv sX note if 43 like was said originally we are wrong CI does not contain no way to avoid error completely Now go around the room and get 10 guesses from other people Calculate the mean and standard deviation for your sample 0 8 X 0 6 0 4 s 0 2 0 41 42 43 44 45 Xi n 2 2 i Xi i Xi n n 1 Calculate standard error of the mean s s sX n 10 we have d f n 1 10 1 9 so from the t distribution table we find that tcv 2 262 I ll calculate the population mean for the class Sample Mean each of you will calculate a confidence interval for your sample with 0 05 10 11 12 CONFIDENCE INTERVALS WHAT DOES THIS MEAN CONFIDENCE thus we conclude with 95 confidence that your interval contains we never say that the confidence interval contains with probability 0 95 CI95 X tcv sX CI95 X 2 262 sX CI95 this is a probabilistic statement about the interval is a parameter a fixed number either the interval contains or it does not we can say that the procedure of producing CI s produce intervals that contain with probability 0 95 we do talk about the confidence that an interval includes different samples produce different confidence intervals but 95 of the time the interval would contain we would say that the confidence interval contains with confidence of 0 95 check the confidence is in the procedure of calculating CIs 13 14 15 CENSUS DEBATE CONCLUSIONS NEXT TIME the US Constitution mandates that the government take a full census every 10 years estimation more on estimation confidence intervals with 250 million people this is difficult normal distribution t distribution relationship between confidence intervals and hypothesis testing especially for certain groups of people Hispanics working mothers interpretation statistical precision Less than 5 of published psychological data is wrong census information is used to guide Federal aid billions of dollars census bureau wants to use statistical estimation to complement the census especially for groups that they know get missed in the straight head count congress will not allow it 16 17 18