- On p. 435 in the paragraph below Example 6.1, the authors
say what it means for the sample mean to be an unbiased
estimator of the population mean. How do they put it?
They say that the sample mean has no systematic
tendancy to over or underestimate the population mean
(over repeated trials).
-
Determine the following probabilities for the standard normal
distribution:
- P(-1.645 < Z < 1.645)
- P(-1.960 < Z < 1.960)
- P(-2.576 < Z < 2.576)
How should the values of these probabilities be related
to the values of C found in the table on p. 439?
Your answers should match these values of C exactly.
That's because 90% of values along the standard normal curve
lie within 1.645 standard deviations from the center, 95%
within 1.96 s.d.'s, and 99% within 2.576 s.d.'s.
-
The authors indicate that it is unrealistic, in practice,
for one to know the value of the population's standard deviation
s. For now, however,
the problems assigned will provide
this information as if it can be known. Assuming you have it,
how do you determine the margin of error corresponding
to a given level of confidence for the mean of a sample?
You must determine the critical value z* (say, using
Table D) that goes with your confidence level, multiply this
number by the population standard deviation s and divide
by the square root of n.
-
Just what is a 99% confidence interval? Apply your interpretation
to the scenario in which a television news organization claims
that Candidate A is projected to carry the State of Michigan
with 95% confidence 56%±3%.
A 99% confidence interval should be interpreted as saying that
if samples of the same size were taken repeatedly and a 99%
confidence interval found for each, in the long run 99% of
these intervals would contain the true parameter. Another way
of saying this is that the probability that our process will
yield a 99% confidence interval that contains the true
parameter is 0.99.
In the election scenario described above, the news organization
is not saying with certainty that Candidate A has won Michigan.
It is saying that, assuming samples were suitably random, the
probability that the true proportion of votes received by
Candidate A lies between 53% and 59% is 0.95.
-
Why do each of the three bulleted items on p. 442 decrease
the margin of error?
- Lowering the level of confidence C correspondingly
lowers the value of z*, meaning that we are
looking at values that extend to fewer standard deviations
away from our sample statistic.
- A larger sample size gives us less variability for our
sampling statistic (i.e., smaller standard deviation for
the sampling distribution). Though we may hold
z* (i.e., the number of standard deviations away
from our sample statistic) fixed, the size of one standard
deviation decreases.
- A reduction in s corresponds to a reduction in the
standard deviation of our sampling distribution. The
overall effect is much as in the case of the previous
bullet.
-
Read the fine print. On pp. 444-45, you are told just how
trustworthy our inference procedure of finding confidence
itervals is.