WEEK 6 – HOMEWORK 6:
LANE CHAPTERS, 11, 12, AND 13; ILLOWSKY CHAPTERS 9, 10
INTRODUCTION TO
HYPOTHESIS TESTING
WHAT IS A HYPOTHESIS TEST?
Here we are testing claims about the TRUE POPULATION’S STATISTICS based on SAMPLES we have taken.
The most
common statistic of interest is of course the POPULATION MEAN (
µ
).
But, we can also test its VARIANCE and its
STANDARD DEVIATION.
(We can also compare TWO or more means to see if there are significant differences.
We must have a basic hypothesis, referred to as the NULL Hypothesis (Ho) and an ALTERNATE Hypothesis (Ha).
Our
NULL ( and ALTERNATE) Hypotheses
can take three forms:
(1)
Ho:
µ
<
some number;
Ha:
µ
> that number
(<
is “less than or equal to” and >
is “greater than or equal to” ),
(2)
Ho:
µ
>
some number;
Ha:
µ
< that number , or
(3)
Ho:
µ
= some number;
Ha
µ
≠ that number
;
(
≠
means “not equal
NOTE THAT Ho
MUST HAVE THE “EQUALS” IN IT WHEREAS Ha
NEVER DOES.
(1)
Is referred to as a “ONE-TAILED TEST TO THE LEFT”
(2) Is a “ONE-TAILED TEST TO THE RIGHT”
(3) Is a “TWO-TAILED TEST”
NEXT
, we need to decide what level of significance, i.e.(how sure we want to be about our hypothesis.
This is where
comes in again.
Do we want to test at the 10%, 5% or 1% level of significance?
Another wrinkle is that for the TWO-
TAILED test, since our value could be greater OR less than some number, we use
α
/2
for each extreme, so for 10% it’s
5% (0.050) at each end (tail of the curve), for 5% it’s 2.5% (0.0250) at each end, and for 1% it’s 0.5%
(0.0050) at the ends.
You have heard about this kind of split before with confidence intervals, but think about it.
Here is a graphical display of
all this:
to”)
α

As you can see, there is a CRITICAL z-VALUE for each of these test depending on the significance level alpha (
α
) or
α/2.
In HW4 questions 1 and 2, you found the critical z-values for alpha’s of 1%, 5% and 10%, which would work for the one-
tailed tests.
For the two tailed tests we need to split these alphas (α/2) and find the critical z-values (at the positive and
negative tails of the graph) So, for an
α of 1%
(0.0100) it would be α/2 or 0.005 in the left tail (negative z-value)
= -2.575
and for the far right tail (0.005 in that tail) we would have to find the z-value for an area to the LEFT of 99.5% (0.9950)
and this is +z =
+2.575

Continuing on, for an
α
of 5%
for a two-tailed test the z-values for α/2 would correspond to areas under the curve of
0.0250 at each end.
The far left tail would have a negative z-value of
-1.96
(see picture above) and the far right tail
would have a positive z-value of
+1.96
that in the Table represented an area of 97.5% (0.9750) to the LEFT.
Lastly, for an
alpha of 10%
, hence an α/2 at both ends of 5% (the two-tailed test), the negative z-value would be
-1.645
.
The positive z-value marking the upper 5% (Table value from 95% to the left) is
+1.645
.
SO, FOR YOUR USE IN ALL HYPOTHESIS TEST (AND WORKS FOR CONFIDENCE INTERVALS TOO) HERE IS A TABLE OF THE
CRITICAL Z-VALUES FOR THE VARIOUS LEVELS OF SIGNIFICANCE (ALPHA’s) MOST COMMONLY USED.
ALPHA (α)
-Z-value
(LEFT tail)
+Z-value
(RIGHT tail)
ALPHA/2
(α/2)
-Z-value
(LEFT tail)
+Z-value
(RIGHT tail)
1% (0.0100)
-2.33
+2.33
0.5% (0.0050)
-2.575
+2.575
5% (0.0500)
-1.645
+1.645
2.5% (0.0250)
-1.96
+1.96
10% (0.1000)
-1.28
+1.28
5% (0.0500)
-1.645
+1.645