Exploratory Data Analysis
1.3. EDA Techniques
1.3.5. Quantitative Techniques
Interval Estimate for Mean
Confidence limits for the mean
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 two-sided alternative that it does not have a value
the confidence interval is converted to hypothesis-test form.
The test is a one-sample t-test, and it is defined as:
|Confidence Interval Example||
We generated a 95 %, two-sided confidence interval for
data set based on the following information.
N = 195 MEAN = 9.261460 STANDARD DEVIATION = 0.022789 t1-0.025,N-1 = 1.9723 LOWER LIMIT = 9.261460 - 1.9723*0.022789/√195Thus, a 95 % confidence interval for the mean is (9.258242, 9.264679).
We performed a two-sided, one-sample t-test using the
ZARR13.DAT data set
to test the null hypothesis that the population mean is
equal to 5.
H0: μ = 5 Ha: μ ≠ 5 Test statistic: T = 2611.284 Degrees of freedom: ν = 194 Significance level: α = 0.05 Critical value: t1-α/2,ν = 1.9723 Critical region: Reject H0 if |T| > 1.9723
We reject the null hypotheses for our two-tailed t-test because the absolute value of the test statistic is greater than the critical value. If we were to perform an upper, one-tailed test, the critical value would be t1-α,ν = 1.6527, and we would still reject the null hypothesis.
The confidence interval provides an alternative to the hypothesis test. If the confidence interval contains 5, then H0 cannot be rejected. In our example, the confidence interval (9.258242, 9.264679) does not contain 5, indicating that the population mean does not equal 5 at the 0.05 level of significance.
In general, there are three possible alternative hypotheses and rejection regions for the one-sample t-test:
The rejection regions for three posssible alternative hypotheses using our example data are shown in the following graphs.
Confidence limits for the mean can be used to answer the following
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. Both Dataplot code and R code can be used to generate the analyses in this section.|