Transcript in sample (x)

Things you need to know for 9.2
What is the mean and standard deviation for
a binomial random variable?
A binomial distribution is more or less moundshaped and can be reasonably approximated
by the normal distribution as long as….?
Proportions are just another way of looking at
count data. Ex: How many male students I
have in AP Stats vs. the proportion of male
students in class.
9.2 Sample Proportions
Sampling distribution of p-hat
Allows us to find how good p-hat is an
estimate of p
We want to estimate the proportion of
“successes” in the population, so take and
SRS from the population of interest. Our
estimator is
p-hat = count of “successes” in sample (x)
size of sample (n)
What proportion of U.S. teens know that 1492 was
the year in which Columbus “discovered” America? A
Gallup Poll found that 210 out of a random sample of
510 American teens aged 13 to 17 knew this
historically important date.
We use p-hat to gain information about the unknown
population parameter p.
Derived from
rules in Chapter
7
Because the mean
of the sampling
distribution of p-hat
= p, p-hat is an
unbiased estimator
of p.
Std. dev decreases
as n increases.
ROT1: Used throughout the rest of the course
whenever our interest is in drawing a sample to make
inferences about a population.
ROT2: Same conditions for using Normal approx. for
binomial.
Recall from 9.1: The sampling distribution of p-hat is
approximately normal and is closer to normal when n
is large.
The accuracy of the Normal approximation improves
as n increases.
Ex. 9.4
A polling organization asks
an SRS of 1500 first-year
college students whether
they applied to other
colleges. In fact, 35% of all
first-year students applied to
colleges besides the one
they are attending. There are
over 1.7 million first-year
college students.
What is the probability that
the random sample of 1500
students will give a result
within 2 percentage points of
this true value?
Suppose you are going to roll a fair six-sided die 60 times and record p-hat, the
proportion of times that a 1 or a 2 is showing.
1. Where should the distribution of the 60 p-hat values be centered? Justify
your answer.
2. What is the standard deviation of the sampling distribution of p-hat, the
proportion of all rolls of the die that show a 1 or a 2?
3. Describe the shape of the sampling distribution of p-hat. Justify your
answer.
Power companies kill trees growing near their lines to avoid power failures due
to falling limbs in storms. Applying a chemical to slow the growth of the
trees is cheaper than trimming, but the chemical kills some of the trees.
Suppose that one such chemical would kill 20% of sycamore trees. The
power company tests the chemical on 250 sycamores. Consider these an
SRS from the population of all sycamore trees.
4. What are the mean and standard deviation of the proportion of trees that are
killed?
5. What is the probability that at least 60 trees (24% of the sample) are killed?
(Remember to check that you can use the Normal approximation.)