Research Next

### 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.)