Solutions to Selected Exercises 179- How are the median, mean, and mode related for the normal distribution?
For any normal distribution by the symmetry property, the mean and median are
the same, and the distribution is also unimodal with the mode at the mean. So the
three measures are always equal for normal distributions. - How is the t distribution related to the normal distribution? What is different
about the t - statistic particularly when the sample size is small?
Student ’ s t - distribution with n degrees of freedom is approximately the same as a
standard normal distribution when n is large (large is somewhere between 30 and - When n is small, the t - distribution is centered at 0, and is symmetric, but the
tails drop off much more slowly than for the standard normal distribution (small
is from 2 to 30). The smaller the degrees of freedom are, the heavier are the tails
of the distribution. - Assume that the weight of women in the United States who are between the
ages of 20 and 35 years has a normal distribution (approximately), with a
mean of 120 lbs and a standard deviation of 18 lbs. Suppose you could select
a simple random sample of 100 of these women. How many of these women
would you expect to have their weight between 84 and 156 lbs? If the number
is not an integer, round off to the nearest integer.
First, let us compute the Z - statistic. Suppose X is the weight of a girl chosen at
random, then her Z - statistic is ( X − 120)/18. By the assumption that X is normal
or approximately so, Z has a standard normal distribution. We want the probability
P [84 ≤ X ≤ 156]. This is the same as P [(84 − 120)/18 ≤ Z ≤ (156 − 120)/18] = P
[ − 2 ≤ Z ≤ 2 ] = 0.9544. See the table of the standard normal distribution. So the
expected number of women would be 0.9544(100) = 95.44 or 95 rounded to the
nearest integer.
Chapter 5- What are the two most important properties for an estimator?
The most important properties of a point estimator are its bias and variance. These
are the components of the estimator ’ s accuracy. - What is the disadvantage of just providing a point estimate?
As noted in problem 2, accuracy is the most important property of an estimator
and without knowledge or an estimate of the mean square error (or equivalently
the bias and variance) you do not know how good the estimator is. - If a random sample of size n is taken from a population with a distribution
with mean μ and standard deviation σ , what is the standard deviation (or
standard error) of the sample mean equal to?
For a random sample of size n , the sample mean is unbiased and has a standard
deviation of σ n.
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