Transcript The coin is weighted for tails.

Binomial Distribution & Hypothesis Testing:
The Sign Test
Trees Grown
25
Decision
Frequency
20
15
10
No Fertilizer
Fertilizer
a
5
0
49
51
53
Growth (cm)
55
57
The formal steps of hypothesis testing:
Step 1: State your hypotheses.
Step 2: Find the critical value.
Step 3: Calculate the obtained statistic.
Step 4: Make a decision.
Hypotheses: Types and Tails
You must always report TWO hypotheses:
H0: The null hypothesis. (outcome due to chance alone)
H1: The alternative hypothesis. (specific outcome due to IV)
Hypotheses: Types and Tails
Hypotheses can be ONE or TWO-tailed:
This coin is weighted for tails. (directional)
This coin is weighted. (nondirectional)
Example: Two-Tailed. “I think this coin is weighted”
H0: This coin is not weighted.
H1: This coin is weighted.
Example: One-Tailed. “I think this coin is weighted for
tails”
H0: This coin is not weighted for tails.
H1: This coin is weighted for tails.
Example: One-Tailed. “I think this coin is weighted for
heads”
H0: This coin is not weighted for heads.
H1: This coin is weighted for heads.
Example: One-Tailed. “I think this coin is weighted for
heads”
Number of Heads
H0: This coin is not weighted for heads.
H1: This coin is weighted for heads.
The formal steps of hypothesis testing:
Step 1: State your hypotheses.
Step 2: Find the critical value.
Step 3: Calculate the obtained statistic.
Step 4: Make a decision.
The Critical Value of an Inferential Statistic
H0: This coin is not weighted for tails.
H1: This coin is weighted for tails.
?
? ?
Critical Value of the statistic is the value that demarcates our decision about
whether something is normal vs. abnormal. (also, our decision to either support or
provide support against a hypothesis).
An Experiment
How can we determine whether or not our hypothesis is correct?
We need a rule to decide whether or not the probability of obtaining the outcome we
obtained is likely to be due to chance. We need a decision rule.
Alpha level (a): a probability level set by the investigator to
delineate which outcomes will lead to supporting the alternative
hypothesis. Common conventions: .05 or .01
Finding the Critical Value
H0: This coin is not weighted for tails.
H1: This coin is weighted for tails.
a = .05
.0370
.0000
.0148
.0000
.0002
.0046
.0011
Only 15+ tails yields a probability of .05 (without going over).
So the critical value is 15.
Finding the Critical Value
H0: This coin is not weighted for heads.
H1: This coin is weighted for heads.
a = .05
.0370
.0000
.0000
.0002
.0148
.0046
.0011
Only 5 and less tails yields a probability of .05 (without going over).
So the critical value is 5.
Finding the Critical Value
H0: This coin is not weighted.
H1: This coin is weighted.
.0370
.0000
.0000
.0002
.0148
a = .05
.0370
.0000
.0148
.0046
.0011
5 and less or 15+ yields a probability of .05
(without going over). So the critical values are 5 and 15.
.0000
.0002
.0046
.0011
Finding the Critical Value
H0: This coin is not weighted for tails.
H1: This coin is weighted for tails.
a = .01
.0000
.0148
.0000
.0002
.0046
.0011
Only 16+ tails yields a probability of .01 (without going over).
So the critical value is 16.
Finding the Critical Value
H0: This coin is not weighted for heads.
H1: This coin is weighted for heads.
a = .01
.0000
.0000
.0002
.0148
.0046
.0011
Only 4 and less tails yields a probability of .01 (without going over).
So the critical value is 4.
Finding the Critical Value
H0: This coin is not weighted.
H1: This coin is weighted.
a = .01
.0000
.0000
.0000
.0002
.0046
.0011
3 and less or 17+ yields a probability of .01
(without going over). So the critical values are 3 and 17.
.0000
.0002
.0046
.0011
The formal steps of hypothesis testing:
Step 1: State your hypotheses.
Step 2: Find the critical value.
Step 3: Calculate the obtained statistic.
Step 4: Make a decision.
For the sign test, the obtained statistic is the
result of the flip!
The question will usually give you this
information.
For example: I think a coin is weighted for
tails so I flip it 20 times and I get 16 tails.
Therefore, 16 tails is the obtained statistic.
The Critical Value of an Inferential Statistic
a
a and b are probabilities that correspond to different range of
outcomes. They are mutually exclusive and exhaustive.
• As we increase alpha then beta must decrease and vice versa
• As we change alpha we also change the critical value & vice versa.
The formal steps of hypothesis testing:
Step 1: State your hypotheses.
Step 2: Find the critical value.
Step 3: Calculate the obtained statistic.
Step 4: Make a decision.
In Statistics, we do NOT prove ourselves right.
It is not appropriate to say “the alternative hypothesis was right.”
Instead, we use very specific terminology:
Everything is relative to the null hypothesis:
Two types of decisions or outcomes:
RETAIN the null hypothesis.
REJECT the null hypothesis.
The Critical Value of an Inferential Statistic
a
“rejection
region”
a and b are probabilities that correspond to different range of
outcomes. They are mutually exclusive and exhaustive.
• As we increase alpha then beta must decrease and vice versa
• As we change alpha we also change the critical value & vice versa.
The Critical Value of an Inferential Statistic
Retain
The Null
Conclude that the coin
is not weighted for
tails.
Reject
the Null
Conclude that the coin
is indeed weighted for
tails.
a and b are probabilities that correspond to different range of
outcomes. They are mutually exclusive and exhaustive.
• As we increase alpha then beta must decrease and vice versa
• As we change alpha we also change the critical value & vice versa.
A researcher is interested in determining whether or not a coin used to determine
order of play during a basketball game is weighted for tails. He flips the coin 20
times. He flips the coin and gets 16 tails. Test his hypothesis. (use an alpha of .05)
.0370
.0000
.0148
.0000
.0002
.0046
.0011
Step 1: State the null and alternative hypotheses:
H0: The coin is not weighted for tails.
H1: The coin is weighted for tails.
Step 2: Find the critical value.
Only 15+ tails will yield a p-value of .05 (without going over).
A researcher is interested in determining whether or not a coin used to determine
order of play during a basketball game is weighted for tails. He flips the coin 20
times. He flips the coin and gets 16 tails. Test his hypothesis. (use an alpha of .05)
Step 3: Calculate the test statistic:
16 tails
Step 4: Make a decision:
Reject the null hypothesis. The coin is weighted for tails.
Suppose it is your job to test whether a coin used at the start of a hockey
game is weighted. You flip it 12 times and it comes up tails 10 times.
Test the hypothesis that the coin is weighted using an alpha level of .01.
Step 1: State your null and alternative hypotheses:
H0: The coin is not weighted. H1: The coin is weighted.
Step 2: Find the critical value:
.0002
0
.0029
1
.0161
2
.0161
3
4
5
6
7
Number of Tails
8
9
10
.0029.0002
11
12
Step 3: Calculate the obtained statistic:
10
Step 4: Make a decision. We retain the null hypothesis.
.0002
0
.0029
.0029
1
.0002
2
3
4
5
6
7
Number of Tails
8
9
10
11
12
There are 100 students in my Statistics class: 50 females and 50 males
I have a sneaking suspicion that the brightest students are female (sorry
guys!). I would like to test this hypothesis using an alpha level of .05. I
make a list of the students according to their test grade and isolate the top
15 students in the class. 10 of them are females.
.0417
.0139
.0032
0
1
2
3
4
5
6
7
8
9
Number of Females
10
11
12
13
.0005
.0000
14
15
Hypothesis testing with the
Sign Test Analysis
Binomial Distribution 20 events
0.2
PRobability
0.15
0.1
0.05
Number Heads
Decision
Retain H0
Reject H0
State of Reality
H0 is true
H0 is false
Correct Decision
Type II error
Type I error
Correct Decision
21
19
17
15
13
11
9
7
5
3
0
1
A decision rule based on
the probability of the outcome
or a more extreme one being
due to chance runs the risk of
two types of errors.