Download 7.2 Sample Proportions

7.2 Sample Proportions
Objectives
SWBAT:
• FIND the mean and standard deviation of the sampling distribution of a sample
proportion. CHECK the 10% condition before calculating the standard deviation of the
sample proportions.
• DETERMINE if the sampling distribution of sample proportions is approximately Normal.
• If appropriate, use a Normal distribution to CALCULATE probabilities involving a sample
proportion.
In the context of the Candy Machine Applet, explain the difference between the distribution
of the population, the distribution of a sample, and the sampling distribution of the sample
proportion.
• We go to a candy machine filled with Reese’s Pieces. The population proportion of
orange candies is p = 0.45.
• We then draw an SRS of n = 25 candies and see if the sample proportion of orange
candies was close to the actual population proportion of p = 0.45.
• We end up taking a total of 400 samples of 25 candies from the machine.
• The population is all Reese’s Pieces in the machine, with the population proportion of
orange candies p = 0.45.
• The distribution of the sample is each of the 400 SRS’s (of n = 25) along with the 𝑝 of
orange candies from each sample.
• The sampling distribution of the sample proportion is the distribution of all the 𝑝’s from
the SRS’s.
Based on the Candy Machine Applet, describe what we know about the shape, center, and
spread of the sampling distribution of a sample proportion.
How good is the statistic pˆ as an estimate of the parameter p? The
sampling distribution of pˆ answers this question.
Shape : In some cases, the sampling distribution of pˆ can be
approximated by a Normal curve. This seems to depend on both the
sample size n and the population proportion p.
Center : The mean of the distribution is m pˆ = p. This makes sense
because the sample proportion pˆ is an unbiased estimator of p.
Spread : For a specific value of p , the standard deviation s pˆ gets
smaller as n gets larger. The value of s pˆ depends on both n and p.
There is an important connection between the sample proportion p̂ and
the number of "successes" X in the sample.
count of successes in sample X
pˆ =
=
size of sample
n
When is it okay to say that the distribution of 𝑝 is approximately Normal?
• As n increases, the sampling distribution becomes approximately Normal. Before
you perform Normal calculations, check that the Normal condition is satisfied:
np ≥ 10 and n(1 – p) ≥ 10.
What are the mean and the standard deviation of the sampling distribution of a
sample proportion? Are these formulas on the formula sheet? Are there
conditions that need to be met for these formulas to work?
In Chapter 6, we learned that the mean and standard deviation of a
binomial random variable X are
mX = np
sX = np(1 - p)
Since pˆ = X /n = (1/n) × X, we are just multiplying the random variable X
by a constant (1/n) to get the random variable pˆ . Therefore,
1
m pˆ = (np) = p
n
pˆ is an unbiased estimator or p
1
np(1 - p)
s pˆ =
np(1 - p) =
=
2
n
n
p(1 - p)
n
As sample size increases, the spread decreases.
The 10% condition must be met:
These are on the formula sheet.
Example: The superintendent of a large school district wants to know what proportion of
middle school students in her district are planning to attend a four-year college or university.
Suppose that 80% of all middle school students in her district are planning to attend a fouryear college or university. What is the probability that an SRS of size 125 will give a sample
proportion of at most 75%?
Example: The superintendent of a large school district wants to know what proportion of
middle school students in her district are planning to attend a four-year college or university.
Suppose that 80% of all middle school students in her district are planning to attend a fouryear college or university. What is the probability that an SRS of size 125 will give a sample
proportion of at most 75%?
About 8.13% of all SRSs of size 125 will give a sample proportion of middle school students
who are planning to attend a four-year college or university of at most 0.75.