STAT-UB.0103
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Spring 2011
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Homework 7
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1. Based on a sample of 50 x-values having mean 35.36 and standard deviation 4.26, find
a 95% confidence interval for the population mean.
 z 
2. The formula n    /2  is used to select a sample size before doing an
 E 
experiment. The target error is specified as E , and this represents a proposed upper limit
for the absolute error x   . The probability of achieving this limit is specified as
1 – α. The value used for σ is a prior opinion on the standard deviation.
2
Sometimes the experimenter will request an upper bound for H, the half-width of a
confidence interval. For example, if the confidence interval turns out to be 78.2 ± 6.4
then H is 6.4. Suppose that the experimenter specifies H, decides to use a 1 – α
confidence interval, uses σ as a known value, and will use the confidence interval

x ± z / 2
. In terms of H, α, and σ, what value of n would you recommend?
n
3. Suppose that a sample of 200 accounts receivable entries at a large mail-order
business had a mean price of $846.20 and a standard deviation of $1,840.80. Give a
95% confidence interval for the population mean. Be sure to state any assumptions that
you use.
4. Consider the confidence interval for  based on a sample of size n from a
population which is assumed normal.
a.
b.
c.
If n = 10, how much longer (percentagewise) is a 95% confidence interval than
a 90% confidence interval?
If n = 20, how much longer (percentagewise) is a 95% confidence interval than
a 90% confidence interval?
Finding the limiting answer, as n gets very large, to the solution to the questions
above.
5. A sample of 612 voters were asked whether they would vote for candidate Wolczak in
the upcoming election. Of these, 344 said that they intended to vote for Wolczak. Find
the conventional 95% confidence interval for the population parameter p (corresponding
to the proportion who will vote for Wolczak).
6. For the situation of the previous problem, give the 95% continuity-corrected
confidence interval for p.
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STAT-UB.0103





Spring 2011




Homework 7






7. For the situation of the previous two problems, give the 95% Agresti-Coull confidence
interval for p.
8. Hillary had a very busy week in the laboratory. She was asked, over and over again,
to produce 95% confidence intervals. Over this week, she gave 120 such intervals. All
120 problems were independent of each other. What is the probability that eight or more
of her intervals failed to cover the true population parameter?
9. Joe would like to give a confidence interval for the binomial parameter p.
Unfortunately he is only able to take one observation. This observation he’ll call X, and
the only possible outcomes are X = 0 and X = 1. Joe has decided to use these statements:
If X = 0, he’ll say, “I am ____ percent confident that p is in the interval [0, 0.80).”
If X = 1, he’ll say, “I am ____ percent confident that p is in the interval (0.20, 1].”
What number should go in the blank?
HINT: You’ll need a careful definition of the confidence coefficient (the
number in the blank). This should be the minimum value (over all p) of the
probability that the interval is correct. In symbols,
Confidence coefficient = min P interval includes p | p is true value
p
10. Suppose that a sample of ten values from an assumed normal population has mean
x = 210 and standard deviation s = 32. The standard 95% confidence interval for the
s
population mean  is x  t0.025; 9
, which here is (187.1, 232.9).
n
This interval is based on the t distribution for
a.
b.
c.
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n
X 
. It is also true that
s
X 
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P   t0.01;9  n
 t0.04; 9   0.95
(*)
s
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Find the values t0.01; 9 and t0.04; 9 . The first you can get from a printed table.
The second can be obtained from Minitab’s Calc  Probability Distributions
 t  Inverse cumulative probability.
Give in numeric form the 95% confidence interval related to the probability
statement (*) above.
Give two reasons why the standard interval is preferred over (*).
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