i
= 1, ..., n (Xi - µ)2.Virtual Laboratories > Interval Estimation > 1 2 [3] 4 5 6
Suppose that X1, X2, ..., Xn is a random sample from the normal distribution with mean µ and variance d2. In this section we will construct confidence intervals for d2, one of the most important special cases of interval estimation. A parallel section on Tests for the Variance in the Normal Model is in the chapter on Hypothesis Testing.
As usual, we will construct the confidence intervals by finding pivotal variables for d2. The construction depends on whether the mean µ is known or unknown; thus µ is a nuisance parameter for the problem of estimating µ. Finally, recall that the normal family is a location-scale family.
Suppose first that µ is known, although this is usually an artificial assumption in applications. Recall that in this case, the natural estimator of d2 is
W2 =(1 / n) i
= 1, ..., n (Xi - µ)2.
Recall also that V = nW2 / d2 has the chi-square distribution with n degrees of freedom, and hence is a pivotal variable for d2. Now for k > 0 and p in (0, 1), let vk, p denote the quantile of order p for the chi-square distribution with k degrees of freedom. For selected values of k and p and n, vk, p can be obtained from the table of the chi-square distribution or from the quantile applet.
1. Use the pivotal
variable V to show that a 1 - r confidence interval, confidence upper
bound, and confidence lower bound are given as follows:
Note that we have used the equal-tail choice in the construction of the two-sided interval, but the interval is not symmetric about the sample variance W2 (unlike the confidence intervals for µ, which are always symmetric about the sample mean M).
Consider now the more realistic case in which µ, as well as d2, is unknown. In this case, the sample variance is
S2 = [1 / (n - 1)] i
= 1, ..., n (Xi - M)2.
where M = (1 / n) i
= 1, ..., n Xi is the sample mean. Recall that
V = (n - 1)S2 / d2
has the chi-square distribution with n - 1 degrees of freedom, and hence is a pivotal variable for d2.
2. Use the pivotal
variable V to show that a 1 - r confidence interval, confidence upper
bound, and confidence lower bound are given as follows:
3. Use
variance estimation experiment to explore the procedure. Select the
normal distribution. Use various parameter values, confidence levels, sample sizes, and
interval types. For each configuration, run the experiment 1000 times with an update
frequency of 10. As the simulation runs, note that the confidence interval successfully
captures the standard deviation if and only if the value of the pivot variable is between
the quantiles. Note the size and location of the confidence intervals and note how well
the proportion of successful intervals approximates the theoretical confidence level.
One of the key assumptions that we made was that the underlying distribution is normal. Of course, in real statistical problems, we are unlikely to know much about the underlying distribution, let alone whether or not it is normal. Even when the underlying distribution is not normal, the procedures of this section are still used to construct approximate confidence intervals for the variance. You will see in the simulation exercises below that this procedure is not nearly as robust as that of constructing interval estimates for the mean. Nonetheless, if the distribution is not too far from normal, the procedure usually works well.
4. In
variance estimation
experiment, select the gamma distribution. Use
various parameter values, confidence levels, sample sizes, and interval types. For each
configuration, run the experiment 1000 times with an update frequency of 10. Note the size
and location of the confidence intervals and note how well the proportion of successful
intervals approximates the theoretical confidence level.
5. In
variance estimation
experiment, select the uniform distribution. Use
various parameter values, confidence levels, sample sizes, and interval types. For each
configuration, run the experiment 1000 times with an update frequency of 10. Note the size
and location of the confidence intervals and note how well the proportion of successful
intervals approximates the theoretical confidence level.
6. For both
procedures, show that a 1 - a confidence interval, lower bound, and upper bound
for d can be obtained by taking the square root of the corresponding confidence
bounds for d2.
7. Suppose that
the weight of a bag of potato chips (in grams) is a random variable with unknown mean µ
and variance d2. A sample of 75 bags has mean 250 and standard
deviation 10. Construct the 90% confidence interval for d.
8. At a
telemarketing firm, the length of a telephone solicitation (in seconds) is a random
variable with unknown mean µ and variance d2. A sample of 50 calls
has mean length 300 and standard deviation 30. Construct the 95% confidence upper bound
for d.
9.
Using Michelson's
data, construct the 95% two-sided confidence interval, the confidence upper bound, and the
confidence lower bound for the standard deviation of the speed of light in air. Assume that the "true value" is
the known mean..
10.
Using Cavendish's
data, construct the 95% confidence interval, confidence upper bound, and confidence lower bound
for the standard deviation of the density of the earth. Assume that the "true value" is the known mean.
11.
Using Short's
data, construct the 95% two-sided confidence interval, the confidence upper bound, and the
confidence lower bound for the standard deviation of the parallax of the sun. Assume that the "true value" is the
known mean.
12.
For the length of a Sertosa iris petal in Fisher's iris
data,
Construct the 90% confidence interval for d.