CONFIDENCE LIMITS

CONFIDENCE LIMITS

Analysis Command

Generates a confidence interval for the mean.

The confidence interval for the mean is:
\( \bar{x} \pm \frac{t_{(\alpha/2,n-1)} s} {\sqrt{n}} \)

with \( \bar{x} \), s, n, and t denoting the sample mean, the sample standard deviation, the sample size, and the percent point function of the t distribution, respectively.

This confidence interval is based on the assumption that the underlying data is approximately normally distributed. However, this confidence interval is fairly robust against non-normality unless the sample size is small or the departure from normality is severe (in particular, the data is not too skewed).

For lognormally distributed data, the modified Cox method can be used to obtain a confidence interval for the mean. In this case, let Y_i denote the log of X_i where X represents the original data. The confidence interval for the mean is then

\( \bar{Y} + \frac{s^2}{2} \pm t(1-\alpha/2,n-1) \sqrt{\frac{s^2}{n} + \frac{s^4}{2(n-1)}} \)

with \( \bar{Y} \) and \( s \) denoting the mean and standard deviation of Y (i.e., the logged data) and t denotes the t percent point function.

For confidence intervals, computing standard confidence limits for the logged data and then back transforming to obtain confidence limits for the original data does not generate accurate intervals. For sufficiently large samples (based on simulations, Olsson suggests sample sizes larger than 200), using the standard normal based confidence interval should give reasonable results. Using the BOOTSTRAP MEAN PLOT command is an alternative method to obtain the confidence interval for data that is not approximately normally distributed.

<LOWER/UPPER> <LOGNORMAL> CONFIDENCE LIMITS <y>
                        <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
            <LOGNORMAL> is optional;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

If LOWER is specified, a one-sided lower confidence limit is returned. If UPPER is specified, a one-sided upper confidence limit is returned. If neither is specified, a two-sided limit is returned.

If LOGNORMAL is given, the confidence interval based on the lognormal distribution will be used.

This syntax supports matrix arguments for the response variable.

MULTIPLE <LOWER/UPPER> <LOGNORMAL> CONFIDENCE LIMITS
                        <y1> ... <yk>             <SUBSET/EXCEPT/FOR qualification>
where <y1> .... <yk> is a list of 1 to 30 response variables;
            <LOGNORMAL> is optional;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax will generate a confidence interval for each of the response variables. The word MULTIPLOT is optional. That is,

MULTIPLE CONFIDENCE LIMITS Y1 Y2 Y3

is equivalent to

CONFIDENCE LIMITS Y1 Y2 Y3

If LOWER is specified, a one-sided lower confidence limit is returned. If UPPER is specified, a one-sided upper confidence limit is returned. If neither is specified, a two-sided limit is returned.

If LOGNORMAL is given, the confidence interval based on the lognormal distribution will be used.

This syntax supports matrix arguments for the response variables.

REPLICATED <LOWER/UPPER> <LOGNORMAL> CONFIDENCE LIMITS
                        <y> <x1> ... <xk>             <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
            <x1> .... <xk> is a list of 1 to 6 group-id variables;
            <LOGNORMAL> is optional;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax performs a cross-tabulation of the <x1> ... <xk> and generates a confidence interval for each unique combination of the cross-tabulated values. For example, if X1 has 3 levels and X2 has 2 levels, six confidence intervals will be generated.

If LOWER is specified, a one-sided lower confidence limit is returned. If UPPER is specified, a one-sided upper confidence limit is returned. If neither is specified, a two-sided limit is returned.

If LOGNORMAL is given, the confidence interval based on the lognormal distribution will be used.

This syntax does not support matrix arguments.

CONFIDENCE LIMITS Y1
CONFIDENCE LIMITS Y1 SUBSET TAG > 2
MULTIPLE CONFIDENCE LIMITS Y1 TO Y5
REPLICATED CONFIDENCE LIMITS Y X

A table of confidence intervals is printed for alpha levels of 50.0, 75.0, 90.0, 95.0, 99.0, 99.9, 99.99, and 99.999. The sample mean, sample standard deviation, and sample standard deviation of the mean are also printed. The t-value and t-value X standard deviation of the mean are printed in the table. These numbers can be used to construct the equivalent hypothesis test if desired (DATAPLOT does not currently provide a hypothesis test command).

In addition to the CONFIDENCE INTERVALS command, the following commands can also be used:
LET ALPHA = 0.05


LET A = LOWER CONFIDENCE LIMIT Y
LET A = UPPPER CONFIDENCE LIMIT Y
LET A = ONE SIDED LOWER CONFIDENCE LIMIT Y
LET A = ONE SIDED UPPER CONFIDENCE LIMIT Y


The LET ALPHA = command is used to specify the significance level.

In addition to the above LET command, built-in statistics are supported for about 20 different commands (enter HELP STATISTICS for details).

Hypothesis tests for the mean can be performed using the t-test.

None

None

DIFFERENCE OF MEANS CONFIDENCE INTERVAL	= Generate a confidence interval for the difference of two means.
T-TEST	= Perform a t-test.
PREDICTION LIMITS	= Generate a prediction limit.
TOLERANCE LIMITS	= Generate a tolerance limit.

Hahn and Meeker (1991), "Statistical Intervals: A Guide for Practitioners", Wiley, pp. 54-55.

Olsson (2005), "Confidence Intervals for the Mean of a Log-Normal Distribution", Journal of Statistics Education, Vol. 13, No. 1.

Confirmatory Data Analysis

Pre-1987
2010/03: Support for MULTIPLE and REPLICATION options
2010/03: Support for matrix options
2017/07: Support for lognormal confidence limits

 
SKIP 25
READ ZARR13.DAT Y
SET WRITE DECIMALS 5
CONFIDENCE LIMITS Y
    

The following output is generated.

            Confidence Limits for the Mean
                     (Two-Sided)
 
Response Variable: Y
 
Summary Statistics:
Number of Observations:                             195
Sample Mean:                                    9.26146
Sample Standard Deviation:                      0.02278
Sample Standard Deviation of the Mean:          0.00163
 
 
 
-----------------------------------------------------------------
  Confidence       t      t-Value X          Lower          Upper
   Value (%)   Value       SD(Mean)          Limit          Limit
-----------------------------------------------------------------
      50.000   0.675        0.00110        9.26035        9.26256
      75.000   1.153        0.00188        9.25957        9.26334
      90.000   1.652        0.00269        9.25876        9.26415
      95.000   1.972        0.00321        9.25824        9.26467
      99.000   2.601        0.00424        9.25721        9.26570
      99.900   3.341        0.00545        9.25600        9.26691
      99.990   3.973        0.00648        9.25497        9.26794
      99.999   4.536        0.00740        9.25405        9.26886
    

 
SKIP 25
READ GEAR.DAT Y X
SET WRITE DECIMALS 5
REPLICATED CONFIDENCE LIMITS Y X
    

The following output is generated.

            Confidence Limits for the Mean
                     (Two-Sided)
 
Response Variable: Y
Factor Variable 1: X                            1.00000
 
Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99800
Sample Standard Deviation:                      0.00434
Sample Standard Deviation of the Mean:          0.00137
 
 
 
-----------------------------------------------------------------
  Confidence       t      t-Value X          Lower          Upper
   Value (%)   Value       SD(Mean)          Limit          Limit
-----------------------------------------------------------------
      50.000   0.702        0.00096        0.99703        0.99896
      75.000   1.229        0.00169        0.99630        0.99969
      90.000   1.833        0.00251        0.99548        1.00051
      95.000   2.262        0.00310        0.99489        1.00110
      99.000   3.249        0.00446        0.99353        1.00246
      99.900   4.779        0.00656        0.99143        1.00456
      99.990   6.584        0.00904        0.98895        1.00704
      99.999   8.794        0.01208        0.98591        1.01008
 
 
            Confidence Limits for the Mean
                     (Two-Sided)
 
Response Variable: Y
Factor Variable 1: X                            2.00000
 
Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99910
Sample Standard Deviation:                      0.00521
Sample Standard Deviation of the Mean:          0.00164
 
 
 
-----------------------------------------------------------------
  Confidence       t      t-Value X          Lower          Upper
   Value (%)   Value       SD(Mean)          Limit          Limit
-----------------------------------------------------------------
      50.000   0.702        0.00115        0.99794        1.00025
      75.000   1.229        0.00202        0.99707        1.00112
      90.000   1.833        0.00302        0.99607        1.00212
      95.000   2.262        0.00373        0.99536        1.00283
      99.000   3.249        0.00536        0.99373        1.00446
      99.900   4.779        0.00788        0.99121        1.00698
      99.990   6.584        0.01086        0.98823        1.00996
      99.999   8.794        0.01450        0.98459        1.01360
 
 
            Confidence Limits for the Mean
                     (Two-Sided)
 
Response Variable: Y
Factor Variable 1: X                            3.00000
 
Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99540
Sample Standard Deviation:                      0.00397
Sample Standard Deviation of the Mean:          0.00125
 
 
 
-----------------------------------------------------------------
  Confidence       t      t-Value X          Lower          Upper
   Value (%)   Value       SD(Mean)          Limit          Limit
-----------------------------------------------------------------
      50.000   0.702        0.00088        0.99451        0.99628
      75.000   1.229        0.00154        0.99385        0.99694
      90.000   1.833        0.00230        0.99309        0.99770
      95.000   2.262        0.00284        0.99255        0.99824
      99.000   3.249        0.00408        0.99131        0.99948
      99.900   4.779        0.00601        0.98938        1.00141
      99.990   6.584        0.00828        0.98711        1.00368
      99.999   8.794        0.01106        0.98433        1.00646
 
 
            Confidence Limits for the Mean
                     (Two-Sided)
 
Response Variable: Y
Factor Variable 1: X                            4.00000
 
Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99820
Sample Standard Deviation:                      0.00385
Sample Standard Deviation of the Mean:          0.00121
 
 
 
-----------------------------------------------------------------
  Confidence       t      t-Value X          Lower          Upper
   Value (%)   Value       SD(Mean)          Limit          Limit
-----------------------------------------------------------------
      50.000   0.702        0.00085        0.99734        0.99905
      75.000   1.229        0.00149        0.99670        0.99969
      90.000   1.833        0.00223        0.99596        1.00043
      95.000   2.262        0.00275        0.99544        1.00095
      99.000   3.249        0.00395        0.99424        1.00215
      99.900   4.779        0.00582        0.99237        1.00402
      99.990   6.584        0.00802        0.99017        1.00622
      99.999   8.794        0.01071        0.98748        1.00891
 
 
            Confidence Limits for the Mean
                     (Two-Sided)
 
Response Variable: Y
Factor Variable 1: X                            5.00000
 
Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99190
Sample Standard Deviation:                      0.00757
Sample Standard Deviation of the Mean:          0.00239
 
 
 
-----------------------------------------------------------------
  Confidence       t      t-Value X          Lower          Upper
   Value (%)   Value       SD(Mean)          Limit          Limit
-----------------------------------------------------------------
      50.000   0.702        0.00168        0.99021        0.99358
      75.000   1.229        0.00294        0.98895        0.99484
      90.000   1.833        0.00439        0.98750        0.99629
      95.000   2.262        0.00542        0.98647        0.99732
      99.000   3.249        0.00778        0.98411        0.99968
      99.900   4.779        0.01145        0.98044        1.00335
      99.990   6.584        0.01578        0.97611        1.00768
      99.999   8.794        0.02107        0.97082        1.01297
 
 
            Confidence Limits for the Mean
                     (Two-Sided)
 
Response Variable: Y
Factor Variable 1: X                            6.00000
 
Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99879
Sample Standard Deviation:                      0.00988
Sample Standard Deviation of the Mean:          0.00312
 
 
 
-----------------------------------------------------------------
  Confidence       t      t-Value X          Lower          Upper
   Value (%)   Value       SD(Mean)          Limit          Limit
-----------------------------------------------------------------
      50.000   0.702        0.00219        0.99660        1.00099
      75.000   1.229        0.00384        0.99495        1.00264
      90.000   1.833        0.00573        0.99306        1.00453
      95.000   2.262        0.00707        0.99172        1.00587
      99.000   3.249        0.01015        0.98864        1.00895
      99.900   4.779        0.01494        0.98385        1.01374
      99.990   6.584        0.02058        0.97821        1.01938
      99.999   8.794        0.02749        0.97130        1.02629
 
 
            Confidence Limits for the Mean
                     (Two-Sided)
 
Response Variable: Y
Factor Variable 1: X                            7.00000
 
Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    1.00150
Sample Standard Deviation:                      0.00787
Sample Standard Deviation of the Mean:          0.00249
 
 
 
-----------------------------------------------------------------
  Confidence       t      t-Value X          Lower          Upper
   Value (%)   Value       SD(Mean)          Limit          Limit
-----------------------------------------------------------------
      50.000   0.702        0.00175        0.99974        1.00325
      75.000   1.229        0.00306        0.99843        1.00456
      90.000   1.833        0.00456        0.99693        1.00606
      95.000   2.262        0.00563        0.99586        1.00713
      99.000   3.249        0.00809        0.99340        1.00959
      99.900   4.779        0.01190        0.98959        1.01340
      99.990   6.584        0.01640        0.98509        1.01790
      99.999   8.794        0.02190        0.97959        1.02340
 
 
            Confidence Limits for the Mean
                     (Two-Sided)
 
Response Variable: Y
Factor Variable 1: X                            8.00000
 
Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    1.00039
Sample Standard Deviation:                      0.00362
Sample Standard Deviation of the Mean:          0.00114
 
 
 
-----------------------------------------------------------------
  Confidence       t      t-Value X          Lower          Upper
   Value (%)   Value       SD(Mean)          Limit          Limit
-----------------------------------------------------------------
      50.000   0.702        0.00080        0.99959        1.00120
      75.000   1.229        0.00141        0.99898        1.00181
      90.000   1.833        0.00210        0.99829        1.00250
      95.000   2.262        0.00259        0.99780        1.00299
      99.000   3.249        0.00372        0.99667        1.00412
      99.900   4.779        0.00548        0.99491        1.00588
      99.990   6.584        0.00755        0.99284        1.00795
      99.999   8.794        0.01008        0.99031        1.01048
 
 
            Confidence Limits for the Mean
                     (Two-Sided)
 
Response Variable: Y
Factor Variable 1: X                            9.00000
 
Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99829
Sample Standard Deviation:                      0.00413
Sample Standard Deviation of the Mean:          0.00130
 
 
 
-----------------------------------------------------------------
  Confidence       t      t-Value X          Lower          Upper
   Value (%)   Value       SD(Mean)          Limit          Limit
-----------------------------------------------------------------
      50.000   0.702        0.00091        0.99738        0.99921
      75.000   1.229        0.00160        0.99669        0.99990
      90.000   1.833        0.00239        0.99590        1.00069
      95.000   2.262        0.00296        0.99533        1.00126
      99.000   3.249        0.00425        0.99404        1.00255
      99.900   4.779        0.00625        0.99204        1.00455
      99.990   6.584        0.00861        0.98968        1.00691
      99.999   8.794        0.01150        0.98679        1.00980
 
 
            Confidence Limits for the Mean
                     (Two-Sided)
 
Response Variable: Y
Factor Variable 1: X                           10.00000
 
Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99479
Sample Standard Deviation:                      0.00532
Sample Standard Deviation of the Mean:          0.00168
 
 
 
-----------------------------------------------------------------
  Confidence       t      t-Value X          Lower          Upper
   Value (%)   Value       SD(Mean)          Limit          Limit
-----------------------------------------------------------------
      50.000   0.702        0.00118        0.99361        0.99598
      75.000   1.229        0.00207        0.99272        0.99687
      90.000   1.833        0.00308        0.99171        0.99788
      95.000   2.262        0.00381        0.99098        0.99861
      99.000   3.249        0.00547        0.98932        1.00027
      99.900   4.779        0.00805        0.98674        1.00285
      99.990   6.584        0.01109        0.98370        1.00589
      99.999   8.794        0.01482        0.97997        1.00962
    

 
SKIP 25
READ LGN2.DAT Y
SET WRITE DECIMALS 4
LOGNORMAL CONFIDENCE LIMITS Y
    

The following output is generated.

            Two-Sided Confidence Limits for the Mean
               (Log-Normal, Modified Cox Method)
 
Response Variable: Y
 
Summary Statistics:
Number of Observations:                  40
Sample Mean (Raw Data):                  274.96250
Sample Standard Deviation (Raw Data):    310.34271
Sample Mean (Log Data):                  5.12720
Correction Term (s*2/2)                  0.50439
Sample Standard Deviation (Log Data):    1.00438
Sample Standard Deviation of the Mean:   0.19562
 
 
 
-----------------------------------------------------------------
  Confidence       t      t-Value X          Lower          Upper
   Value (%)   Value       SD(Mean)          Limit          Limit
-----------------------------------------------------------------
      50.000   0.681        0.13318      244.30086      318.86400
      75.000   1.168        0.22843      222.10554      350.72852
      90.000   1.685        0.32959      200.73651      388.06468
      95.000   2.023        0.39567      187.90017      414.57519
      99.000   2.708        0.52971      164.32892      474.04163
      99.900   3.558        0.69603      139.15044      559.81675
      99.990   4.333        0.84756      119.58387      651.41519
      99.999   5.071        0.99202      103.49946      752.64886