Download Sample Proportions

The Distribution of Sample
Proportions
Section 9.2.1
Starter 9.2.1
• The distribution of I.Q.’s in a certain school
is known to be N(100,12) A randomly
chosen student is found to have an I.Q. of
85.
– What is the z-score of this student?
– Sketch a standard normal curve. Label the
axis and show the z-score
– What was the probability that the student
would have an I.Q. this low or lower?
Today’s Objectives
• Describe the shape, center, and spread of the
sampling distribution of p̂ when a sample of
size n is taken from a population
• Apply two “rules of thumb” to determine whether
the standard deviation formula and the normal
approximation apply
California Standard 16.0
Students know basic facts concerning the relation
between the mean and the standard deviation of a
sampling distribution and the mean and the standard
deviation of the population distribution.
The Sampling Distribution of
p̂
• Shape: The sampling distribution of sample
proportions will be approximately normal
– The larger the sample size, the better the
approximation
• Center: The mean of the sampling distribution is
exactly p (the true population proportion)
• Spread: The standard deviation of the sampling
distribution is
p(1  p)
n
Rules of Thumb
(When do we get to use the normal
approximation?)
1. The formula for standard deviation formula is
valid only when the population is at least 10
times as large as the sample size
2. The normal approximation is valid only if the
expected counts of BOTH success and failure
are at least 10
– As a formula, this is written:
np  10 and n(1  p)  10
Example 9.4
•
1500 college freshmen were asked whether
they applied for admission to any other college
than the one they chose.
• Assume the true nationwide proportion is 35%.
What is the probability that this sample will
give a result within 2% of the true value?
– In other words: If p = .35, find the probability that p̂
is between .33 and .37
1. Check both rules of thumb for validity.
2. If valid, find the probability.
Answer
• The population of all college freshmen is clearly
more than 10 x 1500, so we can use the standard
deviation formula.
• The sample size n is large enough (1500) that
there will certainly be at least 10 in each group, so
we can assume a normal distribution.
• What is the mean of the distribution of p-hat?
• Calculate the standard deviation of the distribution
of p-hat.
(.35)(1  .35)
 pˆ 
 .0123
1500
• Calculate the probability that 0.33<p-hat<0.37:
– Normalcdf(.33,.37,.35,.0123) = .896, so there is about a
90% probability that the sample will be within 2% of the
true value
Example 9.5
• A recent survey of 1500 American adults
contained 9.2% blacks even though blacks
make up about 11% of the population.
Should we suspect undercoverage bias?
• To answer this, find the probability of
getting a sample proportion of at most
9.2% when the true population proportion
is 11%.
• Check rules of thumb and do the math.
Answer
• The population of all blacks is clearly more
than 10 x 1500, so use S.D. formula
• The sample size n is large enough (1500)
that there will certainly be at least 10 in
each group, so use normal distribution
• Calculate the standard deviation of the
statistic
(.11)(1  .11)
 pˆ 
 .00808
1500
P( pˆ  .092)  normalcdf(0,.092,.11,.00808)  .0129
Today’s Objectives
• Describe the shape, center, and spread of the
sampling distribution of p̂ when a sample of
size n is taken from a population
• Apply two “rules of thumb” to determine whether
the standard deviation formula and the normal
approximation apply
California Standard 16.0
Students know basic facts concerning the relation
between the mean and the standard deviation of a
sampling distribution and the mean and the standard
deviation of the population distribution.
Homework
• Read pages 472 - 477
• Do problems 15, 17, 19 - 21
Homework Question 9.15
•
The Gallup Poll once asked a random sample
of 1540 adults, “Do you happen to jog?”
Suppose that in fact 15% of all adults jog.
a) Find the mean and standard deviation of the
proportion p-hat of the sample who jog. (Assume
the sample is an SRS.)
b) Explain why you can use the formula for the
standard deviation of p-hat in this setting.
c) Check that you can use the normal approximation
for the distribution of p-hat.
d) Find the probability that between 13% and 17% of
the sample jog.
e) What sample size would be required to reduce the
standard deviation of the sample proportion to onehalf the value you found in (a)?