Confidence Intervals
The purpose of taking a random sample from a lot or population and computing a statistic, such as the mean
from the data, is to approximate the mean of the population. How well the sample statistic estimates the
underlying population value is always an issue. A confidence interval addresses this issue because it provides a
range of values which is likely to contain the population parameter of interest.
Confidence intervals are constructed at a confidence level, such as 95%, selected by the user. What does this
mean? It means that if the same population is sampled on numerous occasions and interval estimates are made
on each occasion, the resulting intervals would bracket the true population parameter in approximately 95% of
the cases.
Confidence Intervals are a way of taking data from a sample and saying something about the population
from which the sample was drawn.
Our goal is to get to the point where we can read, understand, and write statements like


We are 95% confident that mean caloric intake of infants of low-income mothers receiving WIC
assistance is 80 to 200 kcal per day greater than that of infants of low-income mothers who do
not receive assistance.
We are 95% confident that the mean total cholesterol level resulting from a canola oil diet is
between 9.3 mg/dl less and 17.2 mg/dl more than the mean cholesterol level resulting from a
rice oil diet.
Example 1
A sample of Alzheimer’s patients are tested to assess the amount of time in stage IV sleep. It has been
hypothesized that individuals suffering from Alzheimer’s disease may spend less time per night in the deeper
stages of sleep. Number of minutes spent is Stage IV sleep is recorded for sixty-one patients. The sample
produced a mean of 48 minutes (Sd=14 minutes) of stage IV sleep over a 24 hour period of time. Compute a 95
percent confidence interval for this data. What does this information tell you about a particular individual’s (an
Alzheimer’s patient) stage IV sleep
Example 2
A sample of students from an introductory psychology class was polled regarding the number of hours they
spent studying for the last exam. All students anonymously submitted the number of hours on a 3 by 5 card.
There were 24 individuals in the one section of the course polled. The data was used to make inferences
regarding the other students taking the course. Mean = 10.9
Standard Deviation = 5.6
Compute a 90 percent confidence interval. What does this tell us?
Confidence Interval Worksheet (Means)
Name _______________________
1. Suppose a student measuring the boiling temperature of a certain liquid observes the readings (in degrees
Celsius) 102.5, 101.7, 103.1, 100.9, 100.5, and 102.2 on 6 different samples of the liquid. He calculates the
sample mean to be 101.82. If he knows that the standard deviation for this procedure is 1.2 degrees, what is the
confidence interval for the population mean at a 95% confidence level?
2. A process is known to produce bricks whose weights are normally distributed with standard deviation 0.12
pounds. A random sample of sixteen bricks from today’s output had a mean weight of 4.07 pounds.
a) Find a 99% confidence interval for the mean weight of all bricks produced today.
(b) Without doing the calculations, state whether a 95% confidence interval for the population mean would
be wider than, narrower than, or the same width as that found in (a).
3. A production manager knows that historically, the amounts of impurities in bags of a chemical follow a
normal distribution with a standard deviation of 3.8 grams. A random sample of nine bags of the chemical
yielded the following amounts of impurities in grams:
18.2
(a)
13.7
15.9
17.4
21.8
16.6
12.3
18.8
16.2
Find a 90% confidence interval for the population mean weight of impurities.
4. A random sample of 1,562 undergraduates enrolled in marketing courses was asked to respond on a scale
from one (strongly disagree) to seven (strongly agree) to the statement: “Most advertising insults the
intelligence of the average customer.” The sample mean response was 3.92 and the sample standard deviation
was 1.57.
(a)
Find a 90% confidence interval for the population mean response.
(b) Without doing the calculations, state whether an 80% confidence interval for the population mean
would be wider than, narrower than, or the same as (a).
5. The Cloze readability procedure is designed to measure the effectiveness of a written communication. (A
score of 57 or more on the Cloze test demonstrates adequate understanding of the written material.) A random
sample of 352 certified public accountants was asked to read financial report messages. The sample mean Cloze
score was 60.41 and the sample standard deviation was 11.28. Find a 90% confidence interval for the
population mean score, and comment on your result.
6. A simple random sample of 25 boxes of candy is selected and the contents of each box are weighed. This
leads to a sample mean of 16.3 ounces with a standard deviation of 3.8 ounces. Compute a 90% confidence
interval for the mean weight of the boxes of candy. Interpret the result.
7. Fill in the blanks with on of the following: increases, decreases, or stays the same where E  z * (
a) As the sample size (n) increases, the margin of error (E)
b) As the confidence level (C) increases, the margin of error (E)
c) As the standard deviation (  ) increases, the margin of error (E)

n
).
.
.
_________.
8. A company that produces white bread is concerned about the distribution of the amount of sodium in its
bread. The company takes a simple random sample of 100 slices of bread and computes the sample mean to be
103 milligrams of sodium per slice.
Construct a 99% confidence interval for the unknown mean sodium level assuming that the population
standard deviation is 10 milligrams.
9. You work for a consumer advocate agency and want to find the mean repair cost of a washing machine. In
the past, the standard deviation of the cost of repairs for washing machines has been $17.50. As part of your
study, you randomly select 40 repair costs and find the mean to be $100.00.
Calculate a 90% confidence interval for the population mean.
10. The actual time it takes to cook a ten pound turkey is a normally distributed. Suppose that a random sample
of 19 ten pound turkeys is taken.Given that an average of 2.9 hours and a standard deviation of .24 hours was
found for a sample of 19 turkeys, calculate a 90% confidence interval for the average cooking time of a ten
pound turkey.