Sampling Distribution
Models
CHAPTER 18 PART 3
We use the sample size to determine the standard deviation
for the sampling distribution. Since n is in the denominator of
the formula, the bigger the n, the smaller the standard
deviation.
Read pages 427 & 428
Sampling Distribution Models
Sampling distributions arise because samples vary. The mean or
proportion will depend on the random sample.
We can always simulate a sampling distribution, but the Central Limit
Theorem saves us the trouble.
Don’t confuse the sampling distribution with the distribution of the
sample.
Beware of observations that are not independent. Random samples are
necessary!
Watch out for skewed populations. The CLT assures us that we can use
the Normal model for the sampling distributions, but the sample sizes
will need to be larger if the population is skewed. A rule of thumb is to
have sample size of at least 30.
Example: In a large class of introductory Statistics, the professor has each person toss a
coin 16 times and calculate the proportion of tosses that were heads. The students
report their results, and the professor plots a histogram of these several proportions.
a) What shape would you expect the histogram to be? Why?
Symmetric. The sample size is small, but it is equally probably to get heads as it is to get tails.
b) Where do you expect the histogram to be centered?
0.5 because that is the probability of flipping a coin and landing on heads
c) How much variability would you expect among these proportions?
𝜎𝑝 =
𝑝𝑞
=
𝑛
.5(.5)
= 0.125
16
d) Explain why a Normal model should not be used here.
It does not meet the conditions since np = .5(16) = 8 and therefore np < 10.
Example: In a large class of introductory Statistics, the professor has each person toss a
coin 26 times and calculate the proportion of tosses that were heads.
a) Describe the sampling distribution using the 68-95-99.7 Rule.
Mean = 0.5, SD = 0.1; 68% of sampling proportions are expected to be between 0.4 and 0.6;
95% are expected to be between 0.3 and 0.7; 99.7% are expected to be between 0.2 and 0.8
b) Confirm that you can use a Normal model here.
np = 13, nq = 13, and coin flips are independent
c) The number of tosses is increased to 64. Draw and label the sampling distribution
model.
Mean = 0.5, SD = 0.0625
d) Explain how the sampling distribution changes as the number of tosses increases.
As n increases, the standard deviation decreases. The distribution will still be symmetric, but
less spread out (taller and more narrow).
Investigative Task:
Simulated Coins
Find a partner and find a coin!
Today’s Assignment:
You STILL need to read the chapter!
Add to HW #10 p.434 #16-20, 30