Estimating µ with Large Samples:
An estimate of a population parameter given by a single number is called a point
estimate of that parameter. We use x bar (sample mean) as a point estimate for µ (the
population mean) and s (the sample standard deviation) as a point estimate for σ (the
population standard deviation.
For large samples of size n > 30,
σ≈s
is a good estimate, for most practical purposes.
Before we can start examples, we need to learn about confidence levels. Confidence
levels discuss the reliability of an estimate will be measured by the confidence level.
Finding the Critical Value
Suppose we want a confidence level of c. (See Figure 8-1, text p. 445).
Theoretically, you can choose c to be any value between 0 and 1, but usually c is equal to
0.90, 0.95, or 0.99. In each case, the value zc is the number such that the area under the
standard normal curve falling between -zc and zc is equal to c. The value zc is called the
critical value for a confidence level of c.
The area under the normal curve from -zc to zc is the probability that the
standardized normal variable z lies in that interval. This means that:
P(-zc < z < zc) = c
*** View Example 1 and Guided Exercise 1 on text p. 446 – 447.
Table 8 – 2 (text p. 448) gives some levels of confidence and corresponding critical
values zc.
Error of Estimate
An estimate is not very valuable unless we have some kind of measure of how
“good” it is. Now that we have studied confidence levels and critical values, the
language of probability can give us an idea of the size of the error of estimate by using
the sample mean x bar as an estimate for the population mean.
For a c confidence level, we know:
P(-zc < z < zc) = c
Summary on Error of Estimate
The error of estimate using x bar as a point estimate for µ is the absolute value of
(x bar - µ). In most practical problems, µ is unknown, so the error of estimate is also
unknown. However, the equation from above allows us to compute an error tolerance E,
which serves as a bound on the error of estimate. Using a c% level of confidence, we can
say the point estimate x bar differs from the population mean µ by a maximal error
tolerance of
E = zc_σ_
√n
Since σ ≈ s for large samples we have:
E = zc_σ_
√n
when n > 30
Where E is the maximal error tolerance on the error of estimate for a given confidence
level c (i.e., the absolute value of (x bar - µ) < E with probability c);
zc is the critical value for the confidence level c;
s is the sample standard deviation;
n is the sample size.
Confidence Intervals (Large Samples)
For large samples (n > 30) taken from a distribution that is approximately moundshaped and symmetrical, and for which the population standard deviation σ is unknown, a
c confidence interval for the population mean µ is given by:
x bar – E < µ < x bar + E
Where x bar = sample mean
E = zc_s_
√n
s = sample standard deviation
c = confidence level (0 < c < 1)
zc = critical value for confidence level c (See Table 8 – 2 for frequently used values.)
n = sample size (n > 30)
*** View Example 2 on text p. 451 – 452.
*** View Guided Exercise 2, 3, and 4 on text p. 453 – 454.
When we use samples to estimate the mean of a population, we generate a small
error. However, samples are useful even when it is possible to survey the entire
population because the use of a sample may yield savings of time or effort in collecting
data.
Complete text p. 454 – 459.