Download Take Notes

Name _______________________________
AP STATISTICS CHAPTER 9:
What is the difference between a parameter and a
statistic?
What does the symbol p̂ represent?
Under what conditions is a statistic said to be
UNBIASED?
The SAMPLING DISTRIBUTION of a statistic is
the
taken by the statistic in all possible samples
.
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 9:
What determines the spread (variability) of the
sampling distribution?
The methods we use for evaluating the variability of
a sampling distribution in this course will apply if….
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 9:
Recall that p̂ =
THE SAMPLING DISTRIBUTION OF A SAMPLE
PROPORTION
Choose an SRS of size n from a
population with population proportion p having some
characteristic of interest. Let p̂ be the proportion of
the sample having that characteristic. Then:
1) The mean of the sampling distribution is
2) The standard deviation of the sampling
distribution is given by
RULES OF THUMB (Conditions that need to be
verified before we can proceed):
1) Use this formula for standard deviation when
2) Use the normal approximation when
Example: A polling organization asks an SRS of
1500 first year college students whether they applied
for admission to any other college. In fact, 35% of
all first-year students applied to colleges besides the
one they are attending. What is the probability that
the random sample of 1500 students will give a result
within 2 percentage points of the true value?
1.
STATE: What is our goal? Re-state the
problem using mathematical symbols.
2.
PLAN: Summarize salient facts and conditions.
3.
DO: Perform necessary calculations.
4.
CONCLUDE: Answer the question, in context.
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 9:
Example 2: Suppose one student tossed a coin 200
times and found only 42% heads. Do you believe that
this is likely to happen?
Example 3: Assume that 30% of the students at HH
wear contacts. In a sample of 100 students, what is
the probability that more than 35% of them wear
contacts?
Example 4: About 11% of American adults are black.
Therefore, the proportion of blacks in an SRS of
1500 adults should be close to .11. If a national
sample contains only 9.2% black, should we suspect
that the sampling procedure is somehow underrepresenting blacks?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 9:
Sample Means:
How do you know if a statistic is unbiased?
General Properties of Sample Means:
Suppose that x is the mean of an SRS of size n
drawn from a large population with mean  and
standard deviation  .
Rule 1:
Rule 2:
When are we allowed to use Rule #2?
Rule 3:
EX) The army reports that the
distribution of head circumference
among soldiers is approximately
normal with mean 22.8 inches and standard
deviation of 1.1 inches.
a) What is the probability that a randomly selected
soldier’s head will have a circumference that is
greater than 23.5 inches?
b) What is the probability that a random sample of
five soldiers will have an average head circumference
that is greater than 23.5 inches?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 9:
Example 2:
Suppose a team of biologists has been studying the
Pinedale children’s fishing pond. Let x represent the
length of a single trout taken at random from the
pond. This group of biologists has determined that
the length has a normal distribution with mean of
10.2 inches and standard deviation of 1.4 inches.
What is the probability that a single trout taken at
random from the pond is between 8 and 12 inches
long?
What is the probability that the mean length of five
trout taken at random is between 8 and 12 inches
long?
What sample mean would be at the 95th percentile?
(Assume n = 5)
Example 3
A soft-drink bottler claims that, on average, cans
contain 12 oz of soda. Let x denote the actual
volume of soda in a randomly selected can. Suppose
that x is normally distributed with s = .16 oz. Sixteen
cans are randomly selected and a mean of 12.1 oz is
calculated. What is the probability that the mean of
16 cans will exceed 12.1 oz?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 9:
IF THE POPULATION HAS ANY DISTRIBUTION,
WE MAY EMPLOY THE
.
When n is
the
distribution.
, the sample mean x has
What are the conditions for using the CLT?
Ex: A hot dog manufacturer asserts that one of its
brands has an average fat content of 18 grams per hot
dog. The standard deviation is 1. In a sample of 36
hot dogs, what is the probability that the mean fat
content will be at least 18.4 grams?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 9:
CHAPTER 9 – SUMMARY
SAMPLE PROPORTIONS
Standard deviation – use when the
population is at least 10 times as
large as the sample.
s µp =
p(1 - p)
n
Use the normal approximation
when np  10 and n 1  p   10
p¾¾
® N ( p, s µp )
Then µ
SAMPLE MEANS
Standard deviation - use when the
population is at least 10 times as
large as the sample.
x 

n
If the population is normally
distributed, then x 
 N ( ,

n
)
Central limit theorem: Given any
distribution, if n is large ( n  30 ),
then x 
 N ( ,

n
)
SUMMARY/QUESTIONS TO ASK IN CLASS