Download Chapter 5 - Statistics

Handout 5: Sampling Distribution of the Proportion
Reading Assignment: Chapter 6
Suppose we conduct a binomial experiment with n trials and get successes on X of the trials. Or suppose we measure a categorical variable for a representative sample of n individuals, and X of them have
responses in a certain category. In each case, we can compute the sample proportion, the proportion of
trials resulting in success, or the proportion in the sample with responses in the specified category, as follows:
p̂ =
number of successes
X
=
n
total number of observations in the sample
If we repeated the binomial experiment or collected a new sample, we would probably get a different value
for the sample proportion.
p
Since X is expected to be around n · p with a standard deviation of n · p · (1 − p), then X
is expected to
√
qn
n·p·(1−p)
p·(1−p)
n·p
, or p with a standard deviation of
.
be around n with a standard deviation of
n
n
Suppose that the manager of the local branch of a bank determines that 40% of all depositors have multiple
accounts at the bank. If you select a random sample of 200 depositors, the percentage of depositors that
have multiple accounts should be around 40%, give or take
.
Source: Levine, Krehbiel, and Berenson. Business Statistics: A First Course. 5th ed. New Jersey: Pearson Education, Inc., 2010. 174. Print.
Law of Large Numbers for Sample Proportion
If you toss a fair coin repeatedly, which experiment do you believe would get closer to 50% head, 100 tosses
or 700 tosses?
The larger the number of tosses, the closer we expect to get to 50%. The reason for this is the smaller
standard deviation (as the sample size increases, the standard deviation –).
The Law of Large Numbers for Sample Proportions states that the sample percentage
(or proportion) tends to get closer to the true percentage as the sample size increase.
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Normal Approximation
Recall that for binomial random variables, the normal distribution can be used to calculate approximate
probabilities. In order to calculate those probabilities we were able to standardize it to a z-score by subtracting off the mean and dividing by the standard deviation. Since p̂ = X
n , the sampling distribution of p̂
looks the same as that of a binomial random variable (see Figure 6.1 in the course pack). Therefore, the
sampling distribution of p̂ is approximately normal and can be standardized as well.
Refer back to the local branch bank manger example. The manager determines that 40% of all depositors have multiple accounts at the bank. If you select a random sample of 200 depositors, what is the
probability that the sample proportion of depositors with multiple accounts is less than 0.30. Remember to
convert to a z-score and plot it on standard normal curve before calculating the normal approximation using
normcdf(LOWER, UPPER).
Source: Levine, Krehbiel, and Berenson. Business Statistics: A First Course. 5th ed. New Jersey: Pearson Education, Inc., 2010. 174. Print.
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iClicker Question
Estimating the Population Proportion p
When we actually take random samples, we don’t know the true population proportion. However, we do
how far apart the
q sample proportion and the true population proportion are likely to be. That information
, the standard deviation of p̂. We will use an observed sample proportion ( p̂ ) to
is contained in p·(1−p)
n
estimate the unknown value of the parameter p and estimate the standard deviation of p̂ by using the sample
value p̂ in the formula. This estimated version is called the standard error of p̂.
r
SE(p̂) =
p̂ · (1 − p̂)
n
As sample size increases, the standard error of p̂ decrease proportionally to the square root of the sample size.
Suppose that 5% of U.S. families have a net worth in excess of one million dollars, but 30% of Microsoft’s
employees are millionaires (Harvard Business Review, July-August, 2000). If a sample of 100 of Microsoft’s
employees are randomly selected, what proportion of the sample will be between 25% and 35% millionaires?
Remember to convert to a z-score and plot it on standard normal curve before calculating the normal approximation using normcdf(LOWER, UPPER).
iClicker Question
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