Page 35 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

14 1 Introduction

example) and on appropriate models and/or conditions that the datasets must
satisfy.
Let us now look in more detail what a confidence level really means. Imagine
that in Example 1.2 we were dealing with a random sample extracted from a
population of a very large number of students, attending the course and subject to
an examination under the same conditions. Thus, only one random variable plays a
role here: the student variability in the apprehension of knowledge. Consider,
further, that we wanted to statistically assess the statement “the student
performance is 3 or above”. Denoting by p the probability of the event “the student
performance is 3 or above” we derive from the dataset an estimate of p, known as
point estimate and denoted p ˆ , as follows:

12 +15 +10
ˆ p = = 0.74.
50

The question is how reliable this estimate is. Since the random variable
representing such an estimate (with random samples of 50 students) takes value in
a continuum of values, we know that the probability that the true mean is exactly
that particular value (0.74) is zero. We then loose a bit of our innate and candid
faith in exact numbers, relax our exigency, and move forward to thinking in terms
of intervals around p ˆ (interval estimate). We now ask with which degree of
certainty (confidence level) we can say that the true proportion p of students with
“performance 3 or above” is, for instance, between 0.72 and 0.76, i.e., with a
deviation – or tolerance – of ε = ±0.02 from that estimated proportion?
In order to answer this question one needs to know the so-called sampling
distribution of the following random variable:

P = ( ∑ n i 1= X i n / ) ,
n

where the X i are n independent random variables whose values are 1 in case of
“success” (student performance ≥ 3 in this example) and 0 in case of “failure”.
When the np and n(1–p) quantities are “reasonably large” P n has a distribution
well approximated by the normal distribution with mean equal to p and standard
(
deviation equal to p 1− p) n / . This topic is discussed in detail in Appendices A
and B, where what is meant by “reasonably large” is also presented. For the
moment, it will suffice to say that using the normal distribution approximation
(model), one is able to compute confidence levels for several values of the
tolerance, ε, and sample size, n, as shown in Table 1.6 and displayed in Figure 1.6.
Two important aspects are illustrated in Table 1.6 and Figure 1.6: first, the
confidence level always converges to 1 (absolute certainty) with increasing n;
second, when we want to be more precise in our interval estimates by decreasing
the tolerance, then, for fixed n, we have to lower the confidence levels, i.e.,
simultaneous and arbitrarily good precision and certainty are impossible (some
trade-off is always necessary). In the “jury verdict” analogy it is the same as if one
said the degree of certainty increases with the number of evidential facts (tending