Transcript Chapter 9 Notes

AP Statistics
Chapter 9 Notes
Ch 9 Vocabulary
parameter: a number that describes the
population
 statistic: a number that can be computed from
the sample data without making use of any
unknown parameters.

–
–
–
–
–
–
µ  population mean
 sample mean
σ  population standard deviation
s  sample standard deviation
p population proportion
 sample proportion
Ch 9 Vocab continued
Sampling variability: The value of a
statistic varies in repeated random
sampling.
 Sampling Distribution: The distribution of
the values taken by a statistic in all
possible samples of the same size from
the same population.

– (Very important to understand)
Bias and Variability
A statistic is an unbiased estimator of a
parameter if the mean of its sampling
distribution is equal to the true value of
the parameter being estimated.
 The variability of a statistic is described by
the spread of its sampling distribution.

– Bigger sample size  smaller spread
– Population size does not matter
Sampling Distribution of a Sample
mean (x-bar)
Mean (μ ) = μ
Standard deviation (σ ) =
 Only use if N > 10n

If an SRS of size n is taken from a
population that is Normally distributed,
then the sampling distribution is also
Normal.
Central Limit Theorem

Draw an SRS of size n from any
population with mean μ and standard
deviation σ. When n is large, the
sampling distribution of the sample mean,
is close to the Normal distribution….
Example
Assume IQ scores are Normally distributed
with a mean of 100 and a standard
deviation of 15.
 1. What is the probability of a randomly
selected person having an IQ score of
more than 120?
 2. What is the probability that a random
sample of 7 people will have a mean IQ
score of more than 120?

Example 2
Assume test scores for a large population
have a mean of 72 and standard deviation
of 8.
 1. What is the probability a randomly
selected person has a test score of less
than 70?
 2. Take a random sample of 40 people.
What is the probability their mean score is
less than 70?

Trends to remember

Means of random samples are less
variable than individual observations.

Means of random samples are more
Normal than individual observations.
Sampling Distribution of a Sample
Proportion ( )

Shape: approximately Normal (see Rule on
following slide).
Mean (μ ) = p
 Std Dev (σ ) =

n  size of SRS
 p  population proportion

Rules for Applying formulas

Rule #1: aka Independence Rule

The formula for standard deviation only
applies if the individuals in the sample are
independent. This occurs if the population
is at least 10 times bigger than the
sample. (N > 10n)
Rules for applying formulas

Rule #2: aka Normality Rule
– For proportions, the sampling distribution is
approximately Normal if np > 10 and
n(1-p) > 10
– For means, the sampling distribution is…
 Normal is the population is Normally distributed.
 approximately Normal if the sample size n is large
enough. (We usually say n needs to be > 30).