Methods for a Single Categorical Variable – Confidence Intervals
Up to this point in the semester we’ve been using sample data to judge whether the observed data is
consistent with a statement about a population parameter or not. However, what happens when the
researcher has no idea what the population parameter might be? In this instance, the research would
use sample data to construct an interval, known as a confidence interval, which provides a range of
______________________________ values for the parameter of interest.
Comments:

A confidence interval allows the researcher to ____________________ the population
parameter of interest.

Several methods exist for constructing a confidence interval for a binomial proportion.
However, we will focus on the _________________________ which is the type of interval JMP
computes.
Example: Osteoporosis is a disease defined as very low bone density resulting in high risk of fracture,
hospitalization, and immobilization. Researchers are interested in estimating the true proportion of
postmenopausal women who suffer from osteoporosis. A random sample of 50 postmenopausal
women was taken, and eight of them said they suffer from osteoporosis.
a. Define the population of interest.
b. Define the sample.
c. How big of a sample was taken for the study? This is your sample size, n.
d. What proportion of the sample suffered from osteoporosis? This is your sample statistic, p̂ .
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e. Next, we’ll need the appropriate quantile from the standard normal distribution. The standard
normal distribution is symmetric and bell-shaped. The quantile values given below are
determined by how big of a middle chunk of the distribution is desired due to the specified
confidence level. Whenever quantiles from the standard normal distribution are needed, you’ll
use one of the following values:
Confidence Level
90%
95%
99%
z quantile
Suppose we are interested in constructing a 95% confidence interval for the true proportion of
menopausal women who suffer from osteoporosis. What z quantile should we use?
f.
Now, we can use the information from parts c, d, and e to construct the confidence interval
using the formula given below.
 2npˆ + z   z
2n + z 
2
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z 2  4npˆ 1 - pˆ 
2  n + z2 
Note: This gives us a range of values that is close to the “middle 95%” of the binomial
distribution with n = 50 and p = 8/50 = 0.16.
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g. Lastly, we must interpret what the interval means.
Calculating the score confidence interval in JMP
First, we’ll need to enter the data into JMP as follows.
Next, choose Analyze  Distribution from the menus at the top of the screen. Then put Osteoporosis
in the Y, Columns box and Count into the Freq box as shown below.
Click OK and you should get the following output.
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Next, click on the little red arrow next to Osteoporosis and choose Confidence Interval  0.95. You
should then get the following output.
Margin of Error
The margin of error is defined as the distance between the center of the confidence interval and either
endpoint. For this problem we have:
Upper Endpoint – Center of Interval = 0.28 – 0.18
So, the resulting margin of error = 0.10.
Questions:
1. What happens to the margin of error when the confidence level decreases? For example, what
will happen if we construct a 90% confidence interval instead of a 95% confidence interval?
2. What will happen to the margin of error and confidence interval if we increase the confidence
level?
Example: In 2005, marine biologists of James Cook University in Queensland, Australia, carried out a
study to investigate the proportion of fish that return to their birthplace. They captured adult female
clown fish from a coral reef near Paua New Guinea’s Kimbe Island and then injected them with an
isotope that would be passed on the female’s developing eggs. Since this particular isotope was very
rare in nature, its presence in the offspring was a very effective tag. The offspring then spent a few
weeks in the open ocean, and the researchers returned to the same reef and captured 15 clown fish that
were settled there. Nine of the 15 fish contained the barium isotope; that is, of the 15 sampled fish that
had settled at the reef, 9 had been spawned there.
Source: Science 4 May 2007: Vol. 316 no. 5825, pp. 742 – 744. A link to
the press release regarding this article is given here.
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Research Question – Do the data provide evidence the majority of the juvenile clown fish settled
at the reef were spawned there?
Construct a 99% confidence interval for the true population proportion of clown fish that had been
spawned at the reef.
Questions:
3. Interpret the 99% confidence interval given above.
4. Does this provide evidence that the majority of clown fish at the reef were spawned there?
Explain.
Example: According to a nationwide survey conducted in 200, 60% of parents with young children
condone spanking their child as a regular form of punishment (Tampa Tribune, Oct. 5, 2000). A group of
researchers in child development has noticed a rise in parents using alternative forms of punishments,
such as timeouts, in lieu of spanking, and decided to test their claim. A random sample of 75 parents of
young children was taken and asked whether they condone spanking as a form of regular punishment.
Forty of the 75 parents said they condone spanking.
Research Question – Is there evidence that fewer parents with young children are condoning
spanking as a regular form of punishment?
Questions:
5. Construct a 90% confidence interval by hand for the proportion of parents with young children
who condone spanking as a regular form of punishment.
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6. Interpret the confidence interval constructed in part a.
7. Using JMP, verify the 90% confidence interval computed in part a.
8. In 2000, 60% of parents with small children condoned spanking as a regular form of punishment.
Using the confidence interval constructed in part a, does this provide evidence that fewer
parents with young children are condoning spanking as a regular form of punishment? Explain.
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