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Homework 5
Due Thursday, October 26th
1. Problem 1 in Section 2.10 on page 74, parts c and d.
2. Problem 4 in Section 2.10 on page 75.
3. If the standard deviation of X is 9 times smaller than the standard deviation of X, then how large must
the sample be? (Again, remember to show your work!)
4. Assume the average lifespan of tires is 30,000 miles with a standard deviation of 6,000 miles. Suppose
we take a sample of 100 tires and calculate X .
a) If we assume that the lifespan of tires follows a normal distribution, find the probability that a
particular tire will last more than 30,500 miles.
b) What is the probability that X will be more than 30,500?
c) To calculate part (b), we do not need the assumption that the lifespan of tires follows a normal
distribution. Why?
5. Suppose that 10% of a population is left-handed. In a sample of 5000 individuals, what is the
approximate probability that between 475 and 550 are left-handed?
6. The scores on a particular exam may be approximated by a continuous distribution that has density
f(x) = 63,574,540(x/100)29(1 – (x/100))9 on the range 0 to 100. Using this density, it is possible to find
the expectation and variance for any particular score is E(X) = 75 and V(X) = 45.73171. Suppose 60
students take the exam. Let X be the average of the 60 scores. What is P( X < 77)?
7. The probability is approximately .1 that a person aged 80 will die within 1 year. An insurance
company insures 900 eighty-year-olds. What is the probability that less than 9% of these people will
die in the following year?
8. The distribution of annual salaries of full-time carpenters has a mean equal to $24,000 and a standard
deviation equal to $2,500. The distribution of annual salaries of full-time welders has a mean equal to
$25,000 and a standard deviation equal to $3,000. Suppose we take random samples of 50 carpenters
and 50 welders.
a) What is the probability that, in the samples, the mean salary for the welders exceeds the mean
salary for the carpenters?
b) What is the probability that, in the samples, the mean salaries for the welders and carpenters are
within $500 of one another?
9. On a Sunday afternoon, a random sample of 400 people is taken to estimate p1, the proportion of the
population that watched a hockey game on TV. On the following Sunday, an independent random
sample of 400 people is taken to estimate p2, the proportion of the population who watched a
basketball game on TV. If p1 = .3 and p2 = .4, find the probability that p̂2  p̂1 in our samples.
(i.e. Find P( p̂1  p̂2 > 0).) Notice that if this event actually occurred, the producer would mistakenly
think that more people watched hockey than basketball.
10. Here is one way of showing that S2 is an unbiased estimator of σ2.
Assume X1, X2, …, Xn are independent with common E[Xi] = μ and V[Xi] = σ2. Show that E(S2) = σ2 by
using the actual definition (formula) for the sample standard deviation.
Note: This problem is showing why dividing the sample standard deviation by n – 1
(instead of n) results in an unbiased estimator of σ 2.
Helpful Hints: Follow the outline below.
n
 x
Step 1: Start with the original definition of S2: S2 =
i 1
 x
2
i
n 1
. On homework one, we showed
2
n
 n 
n  x    xi 
xi2  nx 2

i 1
 i 1  which is also equivalent to s2 = i 1
this was equivalent to s2 =
n 1
n n 1
Show that this is true. (i.e. show the last equation is equivalent to one of the other two.)
n
2
i
n
x
Step 2: Take the Expected Value of s2 =
i 1
2
i
 nx 2
n 1
. Remember that you can pull constants out
front, that in general E(W + Y) = E(W) + E(Y), and that X is a random variable.
Step 3: Fill in what the expected values of the pieces from step 2 are and simplify. Remember that
V[Y] = E[Y 2] – (E[Y])2 is true for any random variable Y. Also remember what the Central
Limit Theorem tells us about E[ X ] and V[ X ]. (Remember that you can use algebra tricks
such as if a + b = c, then a = c – b. )