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

1.3.5.2.

Confidence Limits for the Mean

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:
    YBAR +/-  t(alpha/2,N-1)*s/SQRT(N)
where YBAR is the sample mean, s is the sample standard deviation, N is the sample size, alpha is the desired significance level, and t(alpha/2,N-1) is the upper critical value of the t distribution with N - 1 degrees of freedom. Note that the confidence coefficient is 1 - alpha.

From the formula, it is clear that the width of the interval is controlled by two factors:

  1. As N increases, the interval gets narrower from the SQRT(N) term.

    That is, one way to obtain more precise estimates for the mean is to increase the sample size.

  2. The larger the sample standard deviation, the larger the confidence interval. This simply means that noisy data, i.e., data with a large standard deviation, are going to generate wider intervals than data with a smaller standard deviation.
Definition: Hypothesis Test To test whether the population mean has a specific value, u0, against the two-sided alternative that it does not have a value u0, the confidence interval is converted to hypothesis-test form. The test is a one-sample t-test, and it is defined as:
H0: u = u0
Ha: u <> u0
Test Statistic: T = (YBAR - u0)/(s/SQRT(N))
where YBAR, N, and s are defined as above.
Significance Level: alpha. The most commonly used value for alpha is 0.05.
Critical Region: Reject the null hypothesis that the mean is a specified value, u0, if
    T < -t(alpha/2,N-1)
or
    T > t(alpha/2,N-1)
Sample Output for Confidence Interval Dataplot generated the following output for a confidence interval from the ZARR13.DAT data set:

  
                    CONFIDENCE LIMITS FOR MEAN
                            (2-SIDED)
  
           NUMBER OF OBSERVATIONS     =      195
           MEAN                       =    9.261460
           STANDARD DEVIATION         =   0.2278881E-01
           STANDARD DEVIATION OF MEAN =   0.1631940E-02
  
    CONFIDENCE   T     T X SD(MEAN)     LOWER       UPPER
    VALUE (%)  VALUE                    LIMIT       LIMIT
 ---------------------------------------------------------
      50.000   0.676  0.110279E-02   9.26036       9.26256
      75.000   1.154  0.188294E-02   9.25958       9.26334
      90.000   1.653  0.269718E-02   9.25876       9.26416
      95.000   1.972  0.321862E-02   9.25824       9.26468
      99.000   2.601  0.424534E-02   9.25721       9.26571
      99.900   3.341  0.545297E-02   9.25601       9.26691
      99.990   3.973  0.648365E-02   9.25498       9.26794
      99.999   4.536  0.740309E-02   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 - alpha expressed as a percent), the second column lists the t-value (i.e., t(alpha/2,N-1)), the third column lists the t-value times the standard error (the standard error is s/SQRT(N)), 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 one-sample t-test from the ZARR13.DAT data set:

                       T TEST
                     (1-SAMPLE)
                 MU0 =    5.000000
 NULL HYPOTHESIS UNDER TEST--MEAN MU = 5.000000
  
 SAMPLE:
    NUMBER OF OBSERVATIONS      =      195
    MEAN                        =    9.261460
    STANDARD DEVIATION          =   0.2278881E-01
    STANDARD DEVIATION OF MEAN  =   0.1631940E-02
  
 TEST:
    MEAN-MU0                    =    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.
  1. The first section prints the sample statistics used in the computation of the t-test.

  2. The second section prints the t-test statistic value, the degrees of freedom, and the cumulative distribution function (cdf) value of the t-test statistic. The t-test statistic cdf value is an alternative way of expressing the critical value. This cdf value is compared to the acceptance intervals printed in section three. For an upper one-tailed test, the alternative hypothesis acceptance interval is (1 - alpha,1), the alternative hypothesis acceptance interval for a lower one-tailed test is (0,alpha), and the alternative hypothesis acceptance interval for a two-tailed test is (1 - alpha/2,1) or (0,alpha/2). Note that accepting the alternative hypothesis is equivalent to rejecting the null hypothesis.

  3. The third section prints the conclusions for a 95% test since this is the most common case. Results are given in terms of the alternative hypothesis for the two-tailed test and for the one-tailed test in both directions. The alternative hypothesis acceptance interval column is stated in terms of the cdf value printed in section two. The last column specifies whether the alternative hypothesis is accepted or rejected. For a different significance level, the appropriate conclusion can be drawn from the t-test statistic cdf value printed in section two. For example, for a significance level of 0.10, the corresponding alternative hypothesis acceptance intervals are (0,0.05) and (0.95,1), (0, 0.10), and (0.90,1).
Output from other statistical software may look somewhat different from the above output.
Questions Confidence limits for the mean can be used to answer the following questions:
  1. What is a reasonable estimate for the mean?
  2. How much variability is there in the estimate of the mean?
  3. Does a given target value fall within the confidence limits?
Related Techniques Two-Sample T-Test

Confidence intervals for other location estimators such as the median or mid-mean 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 one-sample t-tests are available in just about all general purpose statistical software programs, including Dataplot.
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