Download 11.2 OINT ESTIMATES AND CONFIDENCE INTERVALS Point

11.2 POINT ESTIMATES AND CONFIDENCE INTERVALS
Point Estimates
Suppose we want to estimate the proportion of Americans who approve of
the president. In the previous section we took a random sample of size 1200
from the population and used the proportion of the people in the sample
who approved of the president to estimate the proportion of the people in
the entire country who approve of the president. The proportion of people
in the sample who approve of the president is an example of what is called
a point estimate. It is called a point estimate because it is a single number
that estimates a population parameter, here the proportion of the people in
the country who approve of the president.
Example 11.3
Suppose that out of 1200 people in the sample, 615 approve of the president. The
point estimate of the proportion of the people in the country who approve of the
president is 615/1200 ⳱ 0.5125.
Example 11.4
Suppose that in the example of the 10 containers of sampled water, the sample
average of the 10 observed E. coli bacteria densities is 1500/cm2 . The point estimate
of the density of E. coli bacteria in the swimming pool is 1500/cm2 .
Example 11.5
It was desired to find the population mean and standard deviation of the ages
of the students enrolled at a particular community college. It was impossible to
survey all the students, so a random sample of 100 students was taken and the
mean and the standard deviation of the ages of the students in the sample were
calculated. The sample mean was 20.25 and the sample standard deviation was
2.25. The sample mean age, X ⳱ 20.25, is a point estimate of the population mean
age, ␮ . The sample standard deviation of the ages, S ⳱ 2.25, is a point estimate of
the population standard deviation, ␴ .
In summary, a point estimate for a population parameter of interest
is a statistic computed from the sample. It is believed to be an effective
estimate of the unknown population parameter. Often, what formula to
use for the statistic is obvious. For example, for the population parameter
␮ , which is the mean over a (presumably large) population, it seems that
the sample mean will be a good estimate. However, in more complex
situations the choice of an estimate of a population parameter is not always
clear. Statisticians may find that either they have no idea how to use the
sample data to estimate the population parameter of interest, or they may
in fact have several equally plausible competing estimates to select from.
A relatively simple example of the latter case is the estimation of ␴ 2 . Let n
denote the sample size. Many statisticians would use S2 ⳱ sum(Xi ⫺ X)2 /n,
while many others would use sum(Xi ⫺ X)2 /(n ⫺ 1). Here Xi denotes the
ith observation of the random sample. A solid argument can be made for
either estimate. In fact, a significant amount of statistical theory is devoted
to finding the “best” estimate of a population parameter.
Let’s consider an example in which the formula for a good estimate is
not obvious.
Example 11.6
Suppose the waiting time for a train that goes to the parking area from a particular
terminal at Chicago’s O’Hare International Airport obeys a continuous uniform
distribution on the time interval [0, T], where T is an unknown population parameter. Thus a train always arrives within T minutes, but it is equally likely to arrive
at any moment during this waiting time. Suppose we interview five randomly
selected passengers and find that their waiting times were (in minutes) 1.4, 4.7, 4.2,
5.1, and 2.1. We want to estimate the parameter T. Clearly the sample mean is a poor
choice for an estimate of T. What formula should we use? A widely used technique
for producing an estimate when we have none in mind, called maximum likelihood
estimation, leads us to an estimate that is equal to the maximum of the five observed
times, which is 5.1 minutes. This choice of the maximum of the observed values as
our point estimate is not obvious. Even though it might not be the “best” choice
for an estimate of T, it seems clearly better than X. We will not consider maximum
likelihood estimation in this book.
A point estimate reports one single value that we estimate to be the
true value of the population parameter. However, point estimates have the
important limitation of not informing us how much the estimate is likely
to be in error. Whenever we estimate a population parameter, we lack
total accuracy. Thus our estimate will almost always be different from the
actual population parameter. So the estimate will almost always have some
amount of error associated with it. By simply reporting the point estimate
of a parameter, we have essentially ignored the important issue of the likely
error size associated with the estimate.
Example 11.7
Reconsider Example 11.5. A second random sample, this time of 200 students, was
taken. Suppose, to keep our explanation simple, the point estimate of the population
mean is the same in both cases, namely 20.25. The standard deviation of the second
sample mean would be much lower than the standard deviation of the first mean,
because the second sample is larger. So even though the two point estimates of the
population mean are the same, the second one is surely more accurate, because it is
based on more information—namely, twice as many observations.
Confidence Intervals
To improve on point estimates, statisticians usually report an interval of
values that they believe the parameter is highly likely to lie in. Usually the
point estimate is the middle point of the interval and the endpoints of the
interval communicate the size of the error associated with the estimate (recall
that point estimates ignore this error) and how “confident” we are that the
population parameter is in the interval. The intervals are called confidence
intervals. Typical confidence levels used in practice for confidence intervals
are 90%, 95%, or 99%, with 95% occurring most frequently in applications.
In Examples 11.1 and 11.3, if we are given a 95% confidence interval for the
proportion of the population that approves of the president, which can be
shown to be the interval (0.48, 0.54), we say we are 95% confident that the
population proportion is contained in the confidence interval. In Examples
11.2 and 11.4, given a 90% confidence interval for the density of bacteria
in the swimming pool, (1490, 1510) say, we say we are 90% confident that
the density of bacteria in the swimming pool is contained in the confidence
interval. (We will learn how to calculate confidence intervals for different
population parameters later in the chapter.)
What does it really mean to state a 95% confidence interval for the
unknown population proportion approving of the president? Although
95% sounds impressive, we cannot be satisfied unless we understand what
it means. Theoretically, it means that the probability is 0.95 that such a
confidence interval, which will be random because the sample it is formed
from is random, will contain (surround) the unknown proportion in the
population approving of the president.
Our experimental view of probability based on the five-step method
will help us more clearly and deeply understand what this probability of
0.95 means practically. Just as we do simulations over and over in the
five-step method, imagine that a statistician does the sampling experiment
of Example 11.1 over and over—1000 times, say—and each time computes
a 95% confidence interval from the 1200 sampled people. Now we can
find the experimental probability given by the proportion of the 1000
confidence intervals that actually covers the true fraction of the population
favoring the president. Since 0.95 is the theoretical probability of the interval
containing the population proportion, this experimental probability of the
1000 confidence intervals including the true value will also be close to 0.95.
(Below, we will simulate 100 such 95% confidence intervals and calculate
the experimental confidence interval probability.)
Of course, in a real application there will only be one random sample
and hence only one such confidence interval, such as the (0.48, 0.54)
interval of Examples 11.1 and 11.3. But the statistician obtaining this one
sample knows, because of the experimental probability viewpoint, that this
confidence interval is very likely to be correct in the sense that it contains
the true value (since about 95% of such confidence intervals would cover
the true population proportion). In the case of the (0.48, 0.54) interval of
Examples 11.1 and 11.3, we know it is very likely that the true proportion of
people favoring the president lies between 0.48 and 0.54.
Now in light of this insight into how to interpret the confidence interval
percentage, let’s return to the Key Problem. The St. Louis Post Dispatch
explained the concept of its reported confidence interval this way: “[A 95%
confidence interval] means if the survey were taken 100 times, the results
for the [random] group of respondents would each vary no more than 5.7
percent in either direction from the true population percentage opposing
the stadium [about] 95% of these times.” The Post Dispatch quote is a bit
roundabout and hence forces us to go through a slightly tricky piece of
logic (draw yourself a picture if needed). If the interval, which extends 5.7%
in either direction from its midpoint, indeed varies no more than 5.7% in
either direction from the true population mean about 95% of the time, then
about 95% of these intervals must contain the true population percentage as
desired. The Post Dispatch could have more simply and more directly told its
readers that such an interval can be expected to contain the true population
parameter about 95% of the time.
In other words, just as explained above in the presidential popularity
example, the Post Dispatch is pointing out that if you take 100 random
samples from the same population and calculate the confidence interval for
the population proportion for each sample, about 95% of the confidence
intervals will include the true population proportion. That is, you will be
correct in your claim that the unknown population proportion is in the
interval computed using the sample for about 95 of the 100 samples.
Using our five-step method, we now simulate 100 confidence intervals and determine how many of them contain the true population parameter. Suppose that for the Key Problem the true proportion of people in
the county who are opposed to the stadium is 50% (remember that the parameter value is never known to the statistician). In that case the probability
that a person in the sample will be opposed to the stadium is the same as
the probability of heads seen in flipping a fair coin: 0.5. Our goal is to obtain
100 samples of 301 people and calculate the confidence interval for each of
these 100 samples (we will learn to compute such confidence intervals in
Section 11.6 below). Each sampling is the same as flipping a coin 301 times,
recording the number of heads, and calculating the confidence interval for
the proportion of heads in the sample. We then repeat the process 100 times.
The 100 confidence intervals obtained from this process are represented in
Figure 11.1.
0.65
0.60
0.55
0.50
0.45
0.40
0.35
Figure 11.1
One hundred simulated 95% confidence intervals for the Key Problem
assuming a 50/50 population split.
The line across the middle of the graph represents the true population
proportion, 0.50. The 100 confidence intervals are the vertical lines on
the graph. If the confidence interval covers the 0.50 line, then we say that
the true population proportion is contained in the interval. Likewise, if the
confidence interval does not cover the 0.50 line, then we say that the true
population proportion is not contained in the interval. For the graph of
Figure 11.1, we see that 96 out of the 100 confidence intervals cover the
0.50 line. So for the 100 confidence intervals, 96% of them (about 95%, as
expected) contain the true population parameter, 0.50.
Thus, this example of the five-step method clearly illustrates how we are
to correctly interpret a 95% confidence interval. As already discussed above,
it is a sort of statisticians’ success rate or batting average. If a statistician
constructs 100 95% confidence intervals during a year’s work, then, as
our five-step simulation confirms, we can expect about 95% of them to be
“hits”: cases in which the population parameter is contained in the interval.
Similarly we would expect about 5% of them to be “outs,” or misses: cases in
which the population parameter is not contained in the interval. Compared
with baseball, in which a batting average of 0.300 is considered great, a 0.950
confidence interval coverage rate for statisticians is what is usually required.
Batting 0.300 is never guaranteed in baseball, but in fact a statistician can
guarantee a 95% confidence interval, as we shall see.
Confidence intervals have two basic characteristics that we need to
understand. First, given the same set of data, a 95% confidence interval
is wider than a 90% confidence interval, and a 99% confidence interval is
wider than a 95% confidence interval. Thus, the higher the confidence level
we require, the wider the interval we are forced to accept! Of course, a very
wide interval is of little use to the scientist who has sought statistical advice.
Thus there is no “free lunch” in specifying a 99% confidence instead of a
95% confidence, because the price paid is a wider interval. Here are the
90%, 95%, and 99% confidence intervals for the proportion of people in the
country who approve of the president in Example 11.1:
90%: (0.49, 0.53)
95%: (0.48, 0.54)
99%: (0.47, 0.55)
As you can see from this example, the higher the level of confidence, the
wider the confidence interval needs to be in order to contain the population
proportion with the specified confidence.
The second characteristic of confidence intervals is that, given the
same confidence level, a shorter and hence more informative confidence
interval is associated with more data points. Suppose in the situation of
Example 11.1 a sample of 2400 people was taken from the population and
the number of people in the sample that approved of the president was
1230. The point estimate of the population proportion would be the same:
1230/2400 ⳱ 0.5125. However, a 95% confidence interval for the population
proportion based on this sample of 2400 can be shown to be (0.4925, 0.5325).
This is shorter than the 95% confidence interval for the original sample
of 1200, which can be shown to be (0.4842, 0.5412). Indeed, the result of
increasing the sample size is a shorter interval (of length 0.04 compared with
0.057) for the same confidence level of 95%.
In summary, point estimates provide only a single number to estimate
the value of a population parameter. By contrast, confidence intervals give
a range of values that we reasonably expect will contain the population
parameter. Again, a 95% confidence interval means that if we were to take
a large number of samples (like 100 or 1000) of equal size from the same
population and calculate a confidence interval for the population parameter,
about 95% of the confidence intervals would contain the true value of the
population parameter.
SECTION 11.2 EXERCISES
1. Suppose we want to estimate the proportion
of a city’s residents who drive to work. What
is a good choice for the point estimate of this
proportion?
2. Suppose, instead, we want to estimate the “average” number of miles people living in a city
drive to work. What are two possible choices
for the point estimate of this “average”?
3. Explain the meaning of a 99% confidence
level.
4. Suppose you want a confidence interval for
a population proportion. You want to be as
accurate as possible, so you select a 100% confidence level. What would your confidence
interval have to be?
5. Which confidence interval, when based on
the same data, is wider: an 80% or an 85%
confidence interval?
6. True or false: If you flip a fair coin 100 times,
then calculate a 95% confidence interval, there
is an approximate 95% chance 1/2 will be in
the interval.
7. A Gallup poll of 1013 adults found 61% of
the people in the sample drink alcoholic beverages, which yields a confidence interval of
(58%, 64%). True or false: There is an approximate 95% chance that the percentage of
adults in the population who drink alcoholic
beverages is between 58% and 64%.