1.
Exploratory Data Analysis
1.3. EDA Techniques 1.3.5. Quantitative Techniques


Purpose: Interval Estimate for Mean 
Confidence limits for the mean
(Snedecor and
Cochran, 1989)
are an interval estimate for the
mean. Interval estimates are often desirable because the
estimate of the mean varies from sample to sample. Instead of
a single estimate for the mean, a confidence interval generates
a lower and upper limit for the mean. The interval
estimate gives an indication of how much uncertainty there is
in our estimate of the true mean. The narrower the interval,
the more precise is our estimate.
Confidence limits are expressed in terms of a confidence coefficient. Although the choice of confidence coefficient is somewhat arbitrary, in practice 90%, 95%, and 99% intervals are often used, with 95% being the most commonly used. As a technical note, a 95% confidence interval does not mean that there is a 95% probability that the interval contains the true mean. The interval computed from a given sample either contains the true mean or it does not. Instead, the level of confidence is associated with the method of calculating the interval. The confidence coefficient is simply the proportion of samples of a given size that may be expected to contain the true mean. That is, for a 95% confidence interval, if many samples are collected and the confidence interval computed, in the long run about 95% of these intervals would contain the true mean. 

Definition: Confidence Interval 
Confidence limits are defined as:
From the formula, it is clear that the width of the interval is controlled by two factors:


Definition: Hypothesis Test 
To test whether the population mean has a specific value,
, against
the twosided alternative that it does not have a value
,
the confidence interval is converted to hypothesistest form.
The test is a onesample ttest, and it is defined as:


Sample Output for Confidence Interval 
Dataplot generated the following output for a
confidence interval from the
ZARR13.DAT data set:
CONFIDENCE LIMITS FOR MEAN (2SIDED) NUMBER OF OBSERVATIONS = 195 MEAN = 9.261460 STANDARD DEVIATION = 0.2278881E01 STANDARD DEVIATION OF MEAN = 0.1631940E02 CONFIDENCE T T X SD(MEAN) LOWER UPPER VALUE (%) VALUE LIMIT LIMIT  50.000 0.676 0.110279E02 9.26036 9.26256 75.000 1.154 0.188294E02 9.25958 9.26334 90.000 1.653 0.269718E02 9.25876 9.26416 95.000 1.972 0.321862E02 9.25824 9.26468 99.000 2.601 0.424534E02 9.25721 9.26571 99.900 3.341 0.545297E02 9.25601 9.26691 99.990 3.973 0.648365E02 9.25498 9.26794 99.999 4.536 0.740309E02 9.25406 9.26886 

Interpretation of the Sample Output 
The first few lines print the sample statistics used in calculating
the confidence interval. The table shows the confidence interval
for several different significance levels. The first column lists
the confidence level (which is 1  expressed as a percent),
the second column lists the tvalue (i.e., ), the third column lists
the tvalue times the standard error (the standard error is
), the
fourth column lists the lower confidence limit, and the fifth column
lists the upper confidence limit. For example, for a 95% confidence
interval, we go to the row identified by 95.000 in the first column
and extract an interval of (9.25824, 9.26468) from the last two
columns.
Output from other statistical software may look somewhat different from the above output. 

Sample Output for t Test 
Dataplot generated the following output for a
onesample ttest from the
ZARR13.DAT data set:
T TEST (1SAMPLE) MU0 = 5.000000 NULL HYPOTHESIS UNDER TESTMEAN MU = 5.000000 SAMPLE: NUMBER OF OBSERVATIONS = 195 MEAN = 9.261460 STANDARD DEVIATION = 0.2278881E01 STANDARD DEVIATION OF MEAN = 0.1631940E02 TEST: MEANMU0 = 4.261460 T TEST STATISTIC VALUE = 2611.284 DEGREES OF FREEDOM = 194.0000 T TEST STATISTIC CDF VALUE = 1.000000 ALTERNATIVE ALTERNATIVE ALTERNATIVE HYPOTHESIS HYPOTHESIS HYPOTHESIS ACCEPTANCE INTERVAL CONCLUSION MU <> 5.000000 (0,0.025) (0.975,1) ACCEPT MU < 5.000000 (0,0.05) REJECT MU > 5.000000 (0.95,1) ACCEPT 

Interpretation of Sample Output 
We are testing the hypothesis that the population mean is 5.
The output is divided into three sections.


Questions 
Confidence limits for the mean can be used to answer the following
questions:


Related Techniques 
TwoSample TTest Confidence intervals for other location estimators such as the median or midmean tend to be mathematically difficult or intractable. For these cases, confidence intervals can be obtained using the bootstrap. 

Case Study  Heat flow meter data.  
Software  Confidence limits for the mean and onesample ttests are available in just about all general purpose statistical software programs, including Dataplot. 