Transcript Ch. 3 – Displaying and Describing Categorical Data

Warm Up
• An AP Statistics teacher had 63 students
preparing to take the AP exam. Though they
were obviously not a random sample, he
considered his students to be typical of all
national students. The average score on the
AP exam is a 2.859 with a standard deviation
of 1.324. What’s the probability that his
students will achieve an average score of at
least 3?
Part V – From the Data at Hand to the World at Large
Ch. 18 – Sampling Distribution Models
(Day 3 – The Central Limit Theorem)
Normally Distributed Variables
• Over the last few days, we have solved several
problems involving normally distributed
variables such as height, electric bills, test
scores…
• We were able to use the standard normal (z)
distribution to solve these problems
• What if you were asked to find the probability
that the price of a randomly selected home
fell in a certain range? Could you use z?
• No – since home prices are likely to be skewed
right, the normal distribution wouldn’t work!
Sample Means (again)
• It is the fact that there are a few extreme
values on the right of the distribution of home
prices that stops us from using the Normal
model to represent it
• However, what if we were examining samples
of 20 houses, instead of just individual home
prices – what would this distribution look like?
• The effect of skew in a distribution is lessened
when we look at sample means instead of the
distribution of individual values
The Central Limit Theorem
• Central Limit Theorem: The mean of a
random sample, even when the variable being
measured is not normally distributed, has a
sampling distribution whose shape can be
approximated by a Normal model. The larger
the sample, the better this approximation will
be.
In other words…
• It doesn’t matter whether your original
distribution was normal. The distribution of
sample means will follow the Normal model
anyway, as long as n is large enough.
• How large is large enough?
– This depends on the shape of the original
distribution
– The more “non-normal” the original distribution
is, the larger n has to be
Conditions Revisited
• To use the Normal model to find probabilities involving
sample means, the following conditions must be
present:
1) Randomization: The sample must be selected randomly
2) 10% Condition: The sample size must be less than 10% of
the population size
3) Normal Population or large sample size (n):
– If x follows a normal distribution, the size of the sample
doesn’t matter.
– If x is not normally distributed, then n must be large
enough to make the sampling distribution approximately
normal. We will come back to this idea in a later chapter.
Homework 18-3
• p. 432 # 29, 30, 49