Honors Analysis
Group Members:
Introduction to Confidence Intervals:
Fish Contamination
Nothing like a day boating out on the river! The sun… fresh air… swimming… fishing… and mercury poisoning!
The local health department has hired your group to analyze the fish population to determine the percentage
of fish unsuitable for consumption. (The federal Clean Water Act recommends mercury not exceed 0.3 micrograms
per gram of fish tissue.)
You will be conducting a simulation of this process using Goldfish™ Crackers representing members of the local fish
population. The standard yellow Goldfish will represent fish that are safe to eat, while colored Goldfish are
contaminated.
Since it is impossible to capture and test every fish in your local river, sampling must be used to approximate the
percentage of fish that are unsafe to eat. Keep in mind that when sampling, you will not likely predict the exact
percentage of fish that are contaminated. This activity will guide you through the process of creating a confidence
interval, a range of percentages between which the actual mean of the population will likely lie.
COLLECT YOUR DATA

To take your sample, randomly “catch” ten Goldfish from the class bowl
(try not to look when making your selections so you don’t bias the
selection!). Record the proportion of contaminated fish (as a decimal –
0.1, 0.2, etc.). Repeat four more times to get a sample size of 50.

Calculate the mean proportion of contaminated fish in the five samples.
This value is the sample mean, abbreviated 𝑝̂ (read: “p hat”).
𝑝̂ = ___________
Determine the Confidence Interval

The standard deviation of a sample proportion can be calculated using the formula sd = √
𝑝̂(1−𝑝̂)
, where
𝑛
n
represents the number of items in your sample (50, in this case). What is that value?

Using the 68-95-99.7 rule as a basis, you can create a confidence interval by adding/subtracting standard
deviation values from the sample mean. For example, if your mean is 0.54 and sd = 0.2, then approximately 68%
of sample means can be expected to fall within one standard deviation above or below the mean. So we could
say that the 68% confidence interval for the population parameter lies in the range 0.54 ± 0.1, where 0.1 is
known as the margin of error. In other words, it can be said with 68% confidence that the interval from 0.44 to
0.64 captures the true proportion of contaminated fish in the population. However, 68% certainty is not very
certain… a 95% confidence interval can be calculated within approximately 2 standard deviations of the mean
proportion. Since sd = 0.1, 2 * 0.1 = 0.2, so the 95% confidence interval is 0.54 ± 0.2. In this case, it can be said
with 95% confidence that the mean proportion of contaminated fish falls between 0.34 and 0.74.

Now use your mean proportion and standard deviation to approximate a 95% confidence interval. You may use
𝑝̂(1−𝑝̂)
𝑛
the formula 𝑝̂ ± ∗ Z ∙ √
, where *Z is the number of standard deviations above or below the mean, or use
the description above.
Lower bound: _______________
Upper bound: ______________

Illustrate your confidence interval on the diagram below. (A confidence interval from .12 to .38 is shown as an
example.) Then illustrate the confidence intervals of the other groups in the class.

In actuality, a 95% confidence interval is within 1.96 standard deviations of the mean, not 2 standard deviations.
You can use your z-scores table to find more accurate information. What z-score would correspond to a
confidence interval of 90%? Remember that your table gives the percent of values BELOW a certain point on a
table. You may want to make a sketch of a normal distribution with 10% outside the range evenly divided up in
the left and right sides of the distribution. What z-score would correspond to this percent? (This value is often
denoted *Z). Now calculate a 90% confidence interval for your data.
Lower bound: ____________

Now merge the class data together and create a new 95% confidence interval. You can let *Z = 2 standard
deviations so you can compare to your previous confidence interval. Remember that the value of n will change!
Lower bound: ___________

Upper bound: ____________
Upper bound: ____________
How does the new, larger sample’s confidence interval compare to the smaller sample?
o Are the means different? If so, how?
o
Are the standard deviations different? If so, how?
o
Has the interval changed? If so, how?