Download Unit-22-Basics-of-Hypotheses

Elementary Statistics
Triola, Elementary Statistics 11/e
Unit 22 The Basics of Hypotheses Testing
Hypotheses testing is not all that different from confidence intervals, so let’s do a quick review of
the theory behind the latter.
If it’s our goal to estimate the mean of a population, we’re going to start with the mean of our sample.
We can think of this mean is one of many comprising the sampling distribution, whose mean is 𝝁𝒙̅ = 𝝁
and whose standard deviation is 𝝈𝒙̅ =
𝝈
.
√𝒏
(Review Unit 18, Central Limit Theorem.) Since the sampling
distribution is normally shaped, 95% of the 𝑥̅ ′ 𝑠 are going to fall within ±1.96 standard deviation units
from 𝝁𝒙̅ . (The 95% centered area under the Standard Normal Distribution lies between 𝑧 =
−1.96 𝑎𝑛𝑑 𝑧 = +1.96.) A standard deviation unit for the sampling distribution is 𝝈𝒙̅ =
𝝈
√𝒏
and because
𝝁𝒙̅ = 𝝁, (from the Central Limit Theorem) we have that 95% of the averages, 𝑥,
̅ are going to lie
𝜎
𝜎
√𝑛
between 𝜇 ±1.96 𝑛. Now, if 95% of the 𝑥̅ ′ 𝑠 are within ±1.96
√
𝜎
√𝑛
of 𝜇 then 𝜇 will be within ±1.96
of
95% of all the 𝑥̅ ′ 𝑠 from the sampling distribution. (Think about that). Hence there’s a 95% chance that
𝜎
√𝑛
𝜇 is within ±1.96
of the average, 𝒙̅ of our sample. Now there is one more adjustment we need to
𝜎
√𝑛
make. Since we don’t know 𝜎 we have to replace ±1.96
with 𝑡𝛼⁄2 𝑠⁄√𝑛. . Therefore, our margin of
error E is 𝐸 = 𝑡𝛼⁄2 𝑠⁄√𝑛 and our 95% confidence interval is 𝑥̅ ± 𝑡𝛼⁄2 𝑠⁄√𝑛.
Read the above paragraph over again to really appreciate the beauty of all this.
Recall that we were able to use confidence intervals to verify claims. Let’s suppose that we take a
sample from a population where is claimed that 𝜇 = 25.0., and the 95% confidence interval based on
our sample is (23.75,24.87). Notice that 25.0 is not in the interval. What does that mean? It means
one of two things, either we somehow managed to select a very unusual sample having less than a 5%
chance of being selected, or that the claim is wrong. This is an example of the Rare Event Rule. Which
do you think is more likely, that we picked a “funky” sample, which we only had a 5% chance of doing
so, or the claim is wrong? Most statisticians would agree that most likely the claim is wrong.
This is essentially what we do with hypotheses tests. We test claims. However, with hypotheses tests
we are able to test a greater variety of claims. For example we can test,
a.
b.
c.
d.
e.
The claim is less than some number.
The claim is greater than some number.
The claim is at most some number.
The claim is at least some number
The claim is equal to some number.
These are the same five relations that gave you so much trouble back when you were working with the
Binomial and Poisson distributions. These five relations result in different types of hypotheses testing
involving the “tails” of the Normal or Student t distribution. So first, we have to learn something about
the tails. Consider the following figure,
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Unit 22 The Basics of Hypotheses Testing
The reddish areas are the left and right tails of the distribution. Don’t be concerned with the numbers at
this point. They change with respect to the confidence level. Pictured above is a 95% confidence level
because the area between the two tails is 0.95.
We will be taking raw scores, like 25.25 and converting these scores to t-scores using the following
formula,
𝑡=
𝑥̅ − 𝜇 𝑥̅ − 𝜇
𝑠 = 𝑠 √𝑛
√𝑛
So for example, if 𝑥̅ = 25.25, 𝑠 = 1.32, 𝑛 = 20, 𝜇 = 25.00 (this is the claim), then we have,
𝑡=
x
25.25
25.25 − 25.00
√20 = 0.8470
1.32
s
n
1.32
20
t
0.8470
You can double click the above table, because it is a spreadsheet calculator.
Question #1
Locate approximately this value for t on the graph above. Does it fall under one of the red areas?
Question #2
Now calculate the value of t but instead of using 𝑥̅ = 25.25, use 𝑥̅ = 25.00, the value of the claim.
What did you get?
Locate this value on the graph above. Notice that 𝜇 = 25.00 corresponds to 𝑡 = 0. Now, comes a very
important concept. 𝑥̅ = 25.25 is a distance of 0.8470 units from 𝜇 = 25.00 adjusted for the sample
standard deviation and size, and that distance does not put 25.25 under one of the red areas. If the
̅ and 𝝁 is “too great” then we cannot accept the claim as being true. What is too
distance between 𝒙
great? It’s too great when t ends up being under the red zone. Please reread the above paragraph
several times until it makes sense. It is the central idea of hypotheses testing.
This is the end of Unit 22.
these concepts.
Now turn to MyMathLab to get more practice with
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