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Gifted/Accelerated Math 3
Adapted from Acc Math 3 State Task Unit 1
Sampling Distribution of p̂ (sample proportions)
The Sampling Distribution of a Sample Proportion:
Choose a simple random sample of size n from a large population with population parameter p (proportion
of the population) having some characteristic of interest. Let p̂ be the proportion of the sample having that
characteristic. Then:
o The mean of the sampling distribution is __________.
o The standard deviation of the sampling distribution is ________________.
We would like to know that p̂ is a good estimate for the true proportion of orange Reese’s Pieces. However,
there are guidelines for when we can use the statistic to estimate the parameter.
This leads us to an important result in statistics: the Central Limit Theorem (CLT) for a Sample Proportion:
Choose a simple random sample of size n from a large population with population parameter p having some
characteristic of interest. Then the sampling distribution of the sample proportion p̂ is approximately normal
with mean p and standard deviation
p 1  p 
. This approximation becomes more and more accurate as the
n
sample size n increases, and it is generally considered valid if the population is much larger than the sample,
i.e. np  10 and n(1 – p)  10.
--------------------------------------------------------------------------------------------------------------------------------------Let’s look at the standard deviation a bit more. What happens to the standard deviation as the sample size
increases? Try a few examples to verify your conclusion. Then use the formula to explain why your conjecture
is true.
Sample
Size (n)
5
25
50
100
1000
Let p = 0.25
Let p = 0.73
*If we wanted to cut the standard deviation in half, thus decreasing the variability of p̂ , what would we
need to do in terms of our sample size? (Hint: multiply the formula for standard deviation by ½.)
Caution: We can only use the formula for the standard deviation of p̂ when the population is at least 10
times as large as the sample.
For each of the samples taken yesterday, determine what the population of Reese’s Pieces must be for us to
use the standard deviation formula derived above. Is it safe to assume that the population is at least as large
as these amounts? Explain.
--------------------------------------------------------------------------------------------------------------------------------------A USA Today poll asked a random sample of 1012 U.S. adults what they did with their cereal milk after they
have eaten the cereal. Of the respondents, 67% said that they drink the milk.
a. Is it possible to know for certain what percentage of U.S. adults drink their cereal milk? Explain.
b. Suppose we know that 70% of U.S. adults drink the cereal milk from repreated trials. Find the mean and
standard deviation of the proportion p̂ of the sample that say they drink the cereal milk.
c. Explain why you can use the formula for the standard deviation of p̂ in this situation.
d. What sample size would be required to reduce the standard deviation of the sample proportion to onethird the value you found in part b?
e. Check that you can use the normal approximation for the distribution of p̂ , i.e. the Central Limit
Theorem for Sample Proportions.
f. In Acc. Math II, you learned to calculate z-scores using the following formula: z 
X 
. Note that X

is a statistic,  is a parameter for the mean, and  is the parameter for the standard deviation. More
specifically, whenever we standardize values, we take the estimate (or statistic) minus the corresponding
parameter and divide the difference by the corresponding standard deviation. We will standardize
proportions in the same way. Substitute the statistics and parameters for proportions into the z-score
formula to obtain our standardization formula for proportions.
g. Find the probability of obtaining a sample of 1012 adults in which 67% or fewer say they drink the
cereal milk. (Use the standard normal distribution.)
h. Find the probability that p̂ takes a value between 0.67 and 0.73. This will tell us if a simple random
sample of 1012 adults will usually give a result p̂ within  3 percentage points of the true population
proportion.