Problems: Confidence Intervals Part
1
1. A student takes a random sample of 49 books from the
library and finds that the average number of pages per book
is 270 with a standard deviation of 35.
- a) Construct a 90 percent confidence interval for the
true mean number of pages.
b) Construct a 95% confidence interval for the true
mean.
c) If the student wanted a 95% confidence interval that
was only five pages wide, approximately how big would the
sample size have to be? (Comment: it must be an integer.
You cannot take a sample of 6.345 items.)
2. A student wants to estimate the average age of books
in the library. She goes through the stacks, pulling off a
mix of books that she thinks are representative of the
entire collection. Obtaining 64 books, she finds the average
age of her sample is 29 years and the standard deviation is
16 years.
- a) Are there problems with the way she has taken the
sample? Explain carefully.
- b) If she tries to construct a 95% confidence
interval, what should she get?
- c) If she tries to construct a 99% confidence
interval, what should she get?
- d) Should she be 95% confident of the interval she
found in part b? Explain carefully.
3. A random sample of 100 books is selected from the
library. The average age of the books is 24.3 years and the
standard deviation of the sample is 16 years.
- a) Compute a 95% confidence interval for the true age
of the books in the library.
- b) Compute a 99% confidence interval for the true age
of the books in the library.
- c) A second random sample of 100 books is selected
and 95% and 99% confidence intervals are constructed.
True or false and explain: the second sample will arrive
at exactly the same confidence interval as the
first.
4. A test of 25 felt-tip pens shows that they can write
for an average of 11,291 feet with a standard deviation of
500 feet.
- a) Construct a 95% confidence interval for the
population mean using the normal table.
b) Construct a 95% confidence interval for the population
mean using the t table.
c) Why do your results differ in part a and part b, and
which is the more accurate? Explain clearly.
d) Approximately how much confidence would you have in
the interval 11,291 ± 100?
e) How large must the sample be if we want a precision of
± 100 with 95% confidence?
5. In a random sample of 225 students from a very large
university, only 20% were able to identify Denmark on a map
of Europe. Construct a 95% confidence interval for the
percentage of all students at this university who can
identify Denmark on the map.
6. Below is sample data from a statistical computer
program:
One-Sample Statistics
N: 64
Mean: 118.8910
Std. Deviation: 17.8126
Std. Error Mean: 2.2266
- a) How big is the sample size? How big is the
population?
b) How is the standard error of the mean computed?
c) What is a 95% confidence interval for the true
mean?
d) What is a 99% confidence interval for the true
mean?
7. Suppose that a preliminary sample of 50 items yields a
sample mean price of $60 with a standard deviation of $8.
Approximately how big would the sample size have to be so
that a 95% confidence interval would be the sample mean plus
or minus $1?
8. A student is interested in the amount of hamburger he
gets at a local fast-food eatery. Over a period of two weeks
he obtains a sample of 16 hamburgers and finds that the mean
weight of a hamburger patty is 2.7 ounces with a standard
deviation of .25 ounces.
- a) What is a 95% confidence interval for the
population mean?
- b) Why should you use the Student's t distribution in
this problem instead of the Normal curve?
9. A large lecture class has 210 students. On the last
lecture of the year, a survey is given to all the students
in attendance to see how many read the books. On one book,
112 of the 140 students who handed in the survey indicated
that they read the book and 28 indicate that they did not.
The professor in charge of the class wants a 95% confidence
interval telling how many of the whole class, not just the
sample of 140 students, read the book. Compute the 95%
confidence interval for him, or if it is not possible,
explain why it is not possible. (Hint: is this likely to
satisfy the criteria for a random sample?)
10. You have a box with 100 marbles. Ninety-five are
green and five are red. You pick a marble out of the box but
do not look at it. Would it be better to say, the
probability that the marble in my hand is green is 95%, or,
I am 95% confident that the marble in my hand is green?
(There are t-distribution or t-value calculators on the
Internet that help in solving these problems--that is, they
are easier to use than tables in the back of the book. See,
for example, http://www.tutor-homework.com/statistics_tables/statistics_tables.html.)
Answers here.
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