Control Charts Confidence Interval Student s t test. Quality Control Charts. Confidence Interval. How do we verify accuracy?

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1 Control Charts Confidence Interval Student s t test How do we verify accuracy? Quality Control Charts µ is known or assumed σ is known 3σ action line: 1 in s action 1 in 100 (99%) 2σ warning: 1 in 20 (95%) Confidence Interval The standard deviation indicates precision varies with N How close is experimental mean to µ? Theory allows an estimate of the range within which the true value might fall with a given probability The range is called confidence interval limits of the range are the confidence limits

2 t - statistical factor, depends on N and the confidence level desired confidence limit is product of t and the standard deviation of the mean Values of t given in tables for different confidence levels and degrees of freedom Student s t Table range decreases (N) range increases with required confidence level Example for Confidence limits A soda ash sample is analyzed in the lab. by titation with standard HCl Triplicate determination of Na 2 CO , 93.58, 93.43% Within what range of confidence are you 95% sure that the true value lies?

3 Solution for Confidence Limits mean = 93.5%, s = 0.075% Na 2 CO 3 At 95% confidence and 2 degrees of freedom, t = Confidence limit Notes on Confidence Limits true value falls with range and 93.69% for an infinite number of measurements predict, 95 % confidence, true value falls within two standard deviations for degrees of freedom = infinity, t = 1.96 confidence limits would be close to twice standard deviation of true mean F test Test that two methods have the same standard deviation

4 F test example Below are two sets of replicate analyses on the same sample by two methods. Is standard deviation the same? Your method Standard Method (mean 128) (mean 127) units - mg/dl F test example - calculation Standard Deviation arranged so F>1 Tabulated F for v = 6 and v = 5 is is less than 4.95 so std devs are the sameç Student s t test cf confidence limit equation - limit tends to µ think of µ as the accepted accurate value

5 t test - accepted value known calculate t from N, measured standard deviation, and compare to accepted value if t calc is less then table value then the measuared mean is same as accepted value Example t test (cf accepted) develop method for cu in biological material analysis of NIST biological - same answer? five replicates, mean = 10.8, sd = ±0.7 ppm NIST value Same? 95% confidence? four degrees of freedom (N-1) t=2.776 from table There is an error with 95% confidence Paired t test test compares two sets data / sd for each pooled standard deviation (next slide) replaced with:

6 Pooled Standard Deviation N = N 1 + N etc (N-k) = (N 1-1) + (N 2-1) (N k -1) paired t test assumes both methods have same sd (F test) Example paired t test calc new gravimetric method for iron (III) accuracy checked by determination of iron in an ore sample by new method compared with determination of iron in same ore sample by a standard method results: paired t test iron example Is there a significant difference between the two methods?

7 iron example - F test less than tabulate value (6.26) so two methods have comparable sd, so t test can be applied iron example t test calc tabulated t for 9 degrees of freedom (N1-N2-2) at 95% confidence is 2.262, so there is no statistical difference between the two methods individual differences Multiple t test same as paired t test but for multiple samples of different analyte contents difference between each pair of methods is computed average difference, and sd of differences is used t is calculated: mean of differences

8 Multiple t test example new analyatical method for blood urea nitrogen (BUN) Does the method give the same or a different answer compared to the existing standard method? The two methods have comparable precisions Results of two BUN methods Calculation of t for Multiple Samples tabulated t is at 95% confidence t calc < t table no sig diff between methods

9 Rejection of a Result Q test - rejection quotient If calculate Q > tabulated Q, can reject result Rejection Quotient, Q, at Different Confidence Limits Metals in Air by AA (ng/m 3 ) a Element Authors' Laboratory Clean Room Clean Hood or Bench detection limit c Nov., 1988 June, 1989 Nov., 1988 June, 1989 Nov., 1988 June, 1989 Cu (6) (4) (5) (4) b b 0.02 Fe (6) (4) (3) (4) (4) (4) 0.01 Mn (6) (3) (4) (3) b b Pb (6) (4) (4) (3) b b (4) (4) (4) (4) Sn (6) (4) (4) (3) (4) (3) 0.01 Tl (4) (4) (3) (3) (3) (4)

10 Air Metal Concentrations - footnotes a Data expressed as mean ± standard deviation. Number of replicates (n) in each data set, in parentheses b below detection limit. c Sampling time 6 hours. "detection limit" is smallest concentration that can be measured at 95% confidence. Measurements close to detection limit are unreliable.