Transcript Chapter 18

Chapter 18
Sampling Distribution Models
A Distribution of Sample Proportions
Previously in class our distributions were just
the description of the three characteristics of
data; Shape, Center, Spread.
A distribution of sample proportions will
represent all of the proportions p (probability of
success) from all of the samples of size n.
Modeling the Distribution of Sample
Proportions
Luckily a distribution of sample proportions
mirrors a Normal Model.
A distribution of sample proportions follows the
𝑝𝑞
model N( p, √ ). The mean of a distribution of
𝑛
sample proportions is p, with a standard
𝑝𝑞
deviation of √ .
𝑛
Assumptions and Conditions
We must assume…
1. The sample values are independent of each
other.
2. The sample size, n, must be large enough.
Use these conditions to meet the assumptions.
1. The sample was a SRS, or very near to it.
2. The sample size must be no larger than 10% of
the population.
3. 𝑛𝑝 ≥ 10 𝑎𝑛𝑑 𝑛𝑞 ≥ 10
An example
Lets assume a fair coin is flipped 30 times.
Define success as the coin lands on heads.
1. What is expected value for p?
2. What is the standard deviation for this
sample of proportions?
3. Is it appropriate to use the normal model?
4. What is the probability 𝑝 ≤ 0.54?
Let’s clear a few things up.
p is the proportion (probability of success) for
the population – this is a parameter.
𝑝 is the proportion (probability of success) for a
sample – this is a statistic.
Thus SD(p) is the standard deviation of
proportions of a population while SD(𝑝) is the
standard deviation of a sample of proportions.
Clearing things cont.
You may notice that proportions are actually
representing quantitative data; votes, coin flips,
favorite flavors, etc.
Don’t confuse a sampling distribution with the
distribution of the sample. A sampling
distribution is an imaginary collection of all of
the possible values that a statistic might have
taken for all the random sample.
Practice
• #1, #3,#5 on page 428
• #3
a. 68% between proportions of 0.4 and 0.6,
95% between proportions of 0.4 and 0.7,
99.7% between proportions of 0.2 and 0.8
b. np = 12.5 and nq= 12.5. Both > 10
c. on the board. N(0.5, .0625)
d. Becomes narrower, the standard deviation is
decreasing.
The Sample Distribution Model for the
Mean
If we plot the mean of all of the samples of a
quantitative situation then we have the sample
distribution model for the mean.
For instance if we sampled all of the mean
results of 20 tosses of two fair dice, what would
we expect the distribution to look like?
After 100 tosses? 10,000?
Central Limit Theorem
The mean of a random sample has a sampling
distribution whose shape can be approximated by a
Normal model. The larger the sample, the better
the approximation will be.
Side Notes:
1. Basically, the bigger the sample the more
normal the distribution of means.
2. This is true regardless of the shape of the
population distribution.
Sample Distribution Model for Mean
cont.
We know the that Normal models are specified
by the mean and standard deviation.
1. A sample distribution of means is centered at
the population mean (duh).
2. The standard deviation of means though is
𝜎
𝑆𝐷 𝑦 =
where 𝜎 is the standard
𝑛
deviation of the population and n is the
sample size.
Conditions for CLT
1. The data values must be sampled randomly
or the concept of a sampling distribution
makes no sense.
2. The sample values must be independent.
Always check the 10% condition (sample is
smaller than 10% of population).
3. Large Enough condition. CLT only works
when “n” is large. For our class the test will
be 𝑛 ≥ 30.
#31 on Page 430
a. Find the mean and standard deviation of the
scores.
b. If 40 students are randomly chosen, do we
expect their scores to follow normal model?
c. Consider the mean scores of random samples
of n = 40. Describe the model (shape, center,
and spread).