1/46 Statistics and Data Analysis Professor William Greene Stern School of Business IOMS Department Department of Economics Part 12: Statistical Inference 2/46 Statistics and Data Analysis Part 12 Statistical Inference: Confidence Intervals Part 12: Statistical Inference
3/46 Part 12: Statistical Inference 4/46 Statistical Inference: Point Estimates and Confidence Intervals Statistical Inference Estimation of Population Features Using Sample Data Sampling Distributions of Statisticss
Point Estimates and the Law of Large Numbers Uncertainty in Estimation Interval Estimation Part 12: Statistical Inference 5/46 Application: Credit Modeling 1992 American Express analysis of Application process: Acceptance or rejection
Cardholder behavior Loan default Average monthly expenditure General credit usage/behavior 13,444 applications in November, 1992 Part 12: Statistical Inference 6/46 Modeling Fair Isaacss Acceptance Rate 13,444 Applicants for a Credit Card (November, 1992) Experiment = A
randomly picked application. Let X = 0 if Rejected Let X = 1 if Accepted Rejected Approved Part 12: Statistical Inference 7/46 The Question They Are Really Interested In: Default Of 10,499 people whose application was accepted, 996 (9.49%) defaulted on their credit account (loan). We let X denote the behavior of a credit
card recipient. X = 0 if no default (Bernoulli) X = 1 if default This is a crucial variable for a lender. They spend endless resources trying to learn more about it. Mortgage providers in 2000-2007 could have, but deliberately chose not to. Part 12: Statistical Inference 8/46 The data contained many covariates. Do these help explain the interesting variable? Part 12: Statistical Inference
9/46 Variables Typically Used By Credit Scorers Part 12: Statistical Inference 10/46 Sample Statistics The population has characteristics
Mean, variance Median Percentiles A random sample is a slice of the population Part 12: Statistical Inference 11/46 Populations and Samples Population features of a random variable.
Mean = = expected value of a random variable Standard deviation = = (square root) of expected square root) of expected squared deviation of the random variable from the mean Percentiles such as the median = value that divides the population in half a value such that 50% of the population is below this value
Sample statistics that describe the data Sample mean = x = the average value in the sample Sample standard deviation = s tells us where the sample values will be (square root) of expected using our empirical rule, for example) Sample median helps to locate the sample data on a figure that displays the data, such as a histogram. Part 12: Statistical Inference
12/46 The Overriding Principle in Statistical Inference The characteristics of a random sample will mimic (square root) of expected resemble) those of the population Mean, median, standard deviation, etc. Histogram The resemblance becomes closer as the number of observations in the (square root) of expected random)
sample becomes larger. (square root) of expected The law of large numbers) Part 12: Statistical Inference 13/46 Point Estimation We use sample features to estimate population characteristics. Mean of a sample from the population is an estimate of the mean of the population: x is an estimator of The standard deviation of a sample from the population is an estimator of the standard deviation of the population: s is an estimator of
Part 12: Statistical Inference 14/46 Point Estimator A formula Used with the sample data to estimate a characteristic of the population (square root) of expected a parameter) Provides a single value: Ni1xi x a point estimator of N Ni1(square root) of expected xi x)2 s
a point estimator of N 1 Part 12: Statistical Inference 15/46 Use random samples and basic descriptive statistics. What is the breach rate in a pool of tens of thousands of mortgages? (square root) of expected Breach = improperly underwritten or serviced or otherwise faulty mortgage.) Part 12: Statistical Inference
16/46 The forensic analysis was an examination of statistics from a random sample of 1,500 loans. Part 12: Statistical Inference 17/46 Sampling Distribution The random sample is itself random, since each member is random. Statistics computed from random samples will vary as well. Part 12: Statistical Inference Estimating Fair
Isaacss Acceptance Rate 13,444 Applicants for a Credit Card (November, 1992) Experiment = A randomly picked application. Let X = 0 if Rejected Let X = 1 if Accepted Rejected Approved The 13,444 observations are the population. The true proportion is = 0.780943. We draw samples of N from the 13,444 and use the observations to estimate . 18/46 Part 12: Statistical Inference
19/46 The Estimator The sample proportion we are examining here is a sample mean. X = 0 if the individual's application is rejected X = 1 if the individual's application is accepted 1 N The "acceptance rate" is x = x i. N i1 The population proportion is = 0.780943. x is an estimator of , the population mean. Part 12: Statistical Inference 20/46 x in 100 samples with N = 144 in each sample 0.780943 is the true proportion in the population we are sampling from.
Part 12: Statistical Inference 21/46 The Mean is A Good Estimator Sometimes x is too high, sometimes too low. On average, it seems to be right. The sample mean of the 100 sample estimates is 0.7844 The population mean (true proportion) is 0.7809. Part 12: Statistical Inference 22/46 What Makes it a Good Estimator? The average of the averages will hit
the true mean (square root) of expected on average) The mean is UNBIASED (square root) of expected No moral connotations) Part 12: Statistical Inference 23/46 What Does the Law of Large Numbers Say? The sampling variability in the estimator gets smaller as N gets larger. If N gets large enough, we should hit the
target exactly; The mean is CONSISTENT Part 12: Statistical Inference 24/46 .7 to .88 N=144 .7 to .88 N=1024 .7 to .88 N=4900
Part 12: Statistical Inference Uncertainty in Estimation How to quantify the variability in the proportion estimator --------+--------------------------------------------------------------------Variable| Mean Std.Dev. Minimum Maximum Cases Missing --------+--------------------------------------------------------------------Average of the means of the 100 samples of 144 observations RATES144| .78444 .03278 .715278 .868056 100 0
Average of the means of the 100 samples of 1024 observations RATE1024| .78366 .01293 .754883 .812500 100 0 Average of the means of the 100 samples of 4900 observations RATE4900| .78079 .00461 .770000 .792449 100 0 --------+--------------------------------------------------------------------- 25/46
The population mean (true proportion) is 0.7809. Part 12: Statistical Inference 26/46 Range of Uncertainty The point estimate will be off (square root) of expected high or low) Quantify uncertainty in sampling error. Look ahead: If I draw a sample of 100, what value(square root) of expected s) should I
expect? Based on unbiasedness, I should expect the mean to hit the true value. Based on my empirical rule, the value should be within plus or minus 2 standard deviations 95% of the time. What should I use for the standard deviation? Part 12: Statistical Inference 27/46 Estimating the Variance of the Distribution of Means We will have only one sample! Use what we know about the variance of the mean: Var[mean] = 2/N N 2
Estimate 2 using the data: s2 i1(square root) of expected xi x) N 1 Then, divide s2 by N. Part 12: Statistical Inference 28/46 The Sampling Distribution For sampling from the population and using the sample mean to estimate the population mean:
Expected value of x will equal Standard deviation of x will equal / N CLT suggests a normal distribution Part 12: Statistical Inference 29/46 The sample mean for a given sample may be very close to the true mean The sample mean for a
given sample may be quite far from the true mean This is the sampling variability of the mean as an estimator of Part 12: Statistical Inference 30/46 Recognizing Sampling Variability To describe the distribution of sample means, use the sample x to estimate the population expected value To describe the variability, use the sample standard
deviation, s, divided by the square root of N To accommodate the distribution, use the empirical rule, 95%, 2 standard deviations. Part 12: Statistical Inference 31/46 Estimating the Sampling Variability For one of the samples, the mean was 0.849, s was 0.358. s/N = .0298. If this were my estimate, I would use 0.849 2 x 0.0298 For a different sample, the mean was 0.750, s was 0.433, s/N = .0361. If this were my estimate I would use 0.750 2 x 0.0361
Part 12: Statistical Inference 32/46 Estimates plus and minus two standard errors The interval mean 2 standard errors almost always includes the true value of .7809. The arrows show the cases in which the interval does not contain .7809. Part 12: Statistical Inference How to use these results The sample mean is my best guess of the population mean. I must recognize that there will be estimation error because of random sampling. I use the confidence interval to suggest a range of plausible values for the mean, based on my sample information.
33/46 Part 12: Statistical Inference 34/46 Will the Interval Contain the True Value? Uncertain: The midpoint is random; it may be very high or low, in which case, no. Sometimes it will contain the true value. The degree of certainty depends on the width of the interval.
Very narrow interval: very uncertain. (square root) of expected 1 standard errors) Wide interval: much more certain (square root) of expected 2 standard errors) Extremely wide interval: nearly perfectly certain (square root) of expected 2.5 standard errors) Infinitely wide interval: Absolutely certain. Part 12: Statistical Inference 35/46 The Degree of Certainty The interval is a Confidence Interval The degree of certainty is the degree of confidence. The standard in statistics is 95% certainty (square root) of expected about two standard errors). I can be more confident if I make the interval wider.
I can be 100% confident if I make the interval infinitely wide. This is not helpful. Part 12: Statistical Inference 36/46 67 % and 95% Confidence Intervals Part 12: Statistical Inference 37/46 Monthly Spending Over First 12 Months Population = 10,239 individuals who (1) Received the Card (2) Used the card at
least once (3) Monthly spending no more than 2500. What is the true mean of the population that produced these data? Part 12: Statistical Inference 38/46 Estimating the Mean Given a sample x
= 241.242 S = 276.894 Estimate the population mean Point estimate 241.242 66% confidence interval: 241.242 1 x 276.894/225 = 227.78 to 259.70 95% confidence interval: 241.242 2 x 276.894/225 = 204.32 to 278.162 99% confidence interval: 241.242 2.5 x 276.894/225 = 195.09 to 287.39 N = 225 observations Part 12: Statistical Inference
39/46 Where Did the Interval Widths Come From? Empirical rule of thumb: 2/3 = 66 2/3% is contained in an interval that is the mean plus and minus 1 standard deviation 95% is contained in a 2 standard deviation interval 99% is contained in a 2.5 standard deviation interval. Based exactly on the normal distribution, the exact values would be 0.9675 standard deviations for 2/3 (square root) of expected rather than 1.00) 1.9600 standard deviations for 95% (square root) of expected rather than 2.00) 2.5760 standard deviations for 99% (square root) of expected rather than 2.50) Part 12: Statistical Inference
40/46 Large Samples If the sample is moderately large (square root) of expected over 30), one can use the normal distribution values instead of the empirical rule. The empirical rule is easier to remember. The values will be very close to each other. Part 12: Statistical Inference 41/46 Refinements (Important)
When you have a fairly small sample (square root) of expected under 30) and you have to estimate using s, then both the empirical rule and the normal distribution can be a bit misleading. The interval you are using is a bit too narrow. You will find the appropriate widths for your interval in the t table The values depend on the sample size. (square root) of expected More specifically, on N-1 = the degrees of freedom.) Part 12: Statistical Inference 42/46 Critical Values
For 95% and 99% using a sample of 15: Normal: 1.960 and 2.576 Empirical rule: 2.000 and 2.500 T table: 2.145 and 2.977 Note that the interval based on t is noticeably wider. The values from t converge to the normal values (square root) of expected from above) as N increases. What should you do in practice? Unless the sample is quite small, you can usually rely safely on the empirical rule. If the sample is very small, use the t distribution. Part 12: Statistical Inference
n = N-1 Small sample Large sample 43/46 Part 12: Statistical Inference 44/46 Application
A sports training center is examining the endurance of athletes. A sample of 17 observations on the number of hours for a specific task produces the following sample: 4.86, 6.21, 5.29, 4.11, 6.19, 3.58, 4.38, 4.70, 4.66, 5.64, 3.77, 2.11, 4.81, 3.31, 6.27, 5.02, 6.12 This being a biological measurement, we are confident that the underlying population is normal. Form a 95% confidence interval for the mean of the distribution. The sample mean is 4.766. The sample standard deviation, s, is 1.160. The standard error of the mean is 1.16/17 = 0.281. Since this is a small sample from the normal distribution, we use the critical value from the t distribution with N-1 = 16 degrees of freedom. From the t table (square root) of expected previous page), the value of t[.025,16] is 2.120 The confidence interval is 4.766 2.120(square root) of expected 0.281) = [4.170,5.362]
Part 12: Statistical Inference Application: The Margin of Error The % is a mean of Bernoulli variables, Xi = 1 if the respondent favors the candidate, 0 if not. The % equals 100[(1/652)ixi]. Fundamental polling result: Standard error = SE = sqr[p(square root) of expected 1-p)/N] MOE = 1.96 SE (1) Why do they tell you N=652? (2) What do they mean by MoE = 3.8? (Can you show how they computed it?)
The 95% confidence interval for the proportion of voters who will vote for Clinton is 50% 3.8% = [46.2% to 53.8%] This does not overlap the interval for Trump, so they would predict Clinton to win the election (square root) of expected in NH). The result is not within the margin of error. Aug.6, 2015. http://www.realclearpolitics.com/epolls/2016/president/nh/new_hampshire_trump_vs_clinton-5596.html 45/46 Part 12: Statistical Inference 46/46 Summary
Methodology: Statistical Inference Application to credit scoring Sample statistics as estimators Point estimation Sampling distributions Confidence intervals
Sampling variability The law of large numbers Unbiasedness and consistency Proportion Mean Using the normal and t distributions instead of the empirical rule for the width of the interval. Part 12: Statistical Inference
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