if we live with a deep sense of gratitude, our life will be greatly embellished
DESCRIPTION
If we live with a deep sense of gratitude, our life will be greatly embellished. Hypothesis Test I: Z tests. Logic of hypothesis test Rejection region and p-value Test for one proportion Test for two proportions. Example: New Drug. - PowerPoint PPT PresentationTRANSCRIPT
1
If we live with a deep sense of gratitude, our life will be greatly embellished.
2
Hypothesis Test I: Z tests
Logic of hypothesis testRejection region and p-valueTest for one proportionTest for two proportions
3
Example: New Drug
A pharmaceutical company wants to be able to claim that for its newest medication the proportion of patients who experience side effects is less than 20%.
Q. What are the two possible conclusions (hypotheses) here?
4
The Logic of Hypothesis Test
“Assume Ho is a possible truth until proven false”
Analogical to“Presumed innocent until proven guilty”
The logic of the US judicial system
5
Steps in Hypothesis Test
1. Determine the null (Ho) and alternative (Ha) hypotheses2. Find an appropriate test statistic and pre-set the level of
significance (called level)3. Assuming Ho is true, find rejection region. 4. Reject Ho if the test statistic falls into the rejection
region.5. Report the result in the context of the situation
Alternative steps for 3 & 4:3. Assuming Ho is true, find p-value.4. Reject Ho if p-value < level.
6
Determine Ho and Ha
Ho: nothing is happening (no relationship, no difference,…)
Ha: something is happening (there is a relationship, there is a difference, …)
Rule of Thumb: The “=“ sign must be in Ho. If possible, set what we hope to prove as Ha.
7
Example
p = % of users who will experience side effects
Q. What are Ho and Ha here?
8
Example
Logic: Assume Ho is possibly true until proven false.
Data: 17% of 400 patients who have experienced side effects
How likely is if Ho is true (p>20%)?
If very unlikely reject HoIf not very unlikely cannot reject Ho
%17ˆ p
9
Rejection Region
How extreme (i.e. unlikely) is the observation is too extreme?
Rejection Region is the region when the test statistic falls in, we will “reject Ho”
The rejection region is the most extreme region, determined by the level and the type of Ha
10
Good!Good!(Correct!)(Correct!)
H0 true H0 false
Type II Type II Error, or Error, or ““ Error” Error”
Type I Type I Error, or Error, or ““ Error” Error”
Good!Good!(Correct)(Correct)
we accept H0
we reject H0
Two Types of Errors
Level of Significance
Level of significance also called level is the largest tolerable type I error rate
Example of rejection region for a given level
11
12
P-Value
p-value is the smallest type I error rate if we reject Ho at the observed value
That is, p-value is the probability of seeing as extreme as (or more extreme) what we observe, given Ho is true.
The smaller p-value is, the less likely that what we observe will occur given Ho is true.
That is, smaller p-value means stronger evidence against Ho.
13
P-Value
The level of significance (called level) is usually 0.05
p-value > fail to reject Ho (??) p-value < reject Ho (= accept Ha)
14
Report the Conclusion
Reject Ho: the data shows strong evidence supporting Ha
Eg. The data shows strong evidence that the proportion of users who will experience side effects is less than 20%.
Fail to reject Ho: the data does not provide sufficient evidence supporting Ha
Eg. Based on the data, there is not sufficient evidence to support the proportion is less than 20%
15
Testing Hypotheses about a Proportion
Three possible Ho and Ha
Ho Ha Type
p = po p = po Two-sided
p > po p < po One-sided (lower-tailed)
p < po p > po One-sided (upper-tailed)
Write them all as p=po in the future
16
The z-test for a Proportion
When 1) the sample is a random sample 2) n(po) and n(1-po) are both at least 5,an appropriate test statistic for p is
nppppz
oo
o
)1(ˆ
17
Computing the p-Value for the Z-Test
18
Computing the p-Value for the Z-Test
19
Computing the p-Value for the Z-Test
P-value = P(|Z| > |z*| )= 2 x P(Z > |z*|)
20
Example: New Drug (Conti.)
1. Ho: p > 20% vs. Ha: p < 20%2. Z-test statistic; 3. Find rejection region or p-value4. Decide if reject Ho or not5. Report the conclusion in the context of the
situation
21
Hypothesis Test for the Difference of Two Population Proportions
Step 1. Set up hypothesesHo: p1 = p2 and three possible Ha’s:
Ha: p1 = p2 (two-tailed)or
Ha: p1 < p2 (lower-tailed)or
Ha: p1 > p2 (upper-tailed)
22
Hypothesis Test for the Difference between Two Population Proportions
Step 2. calculate test statistic
where
21
21
11)ˆ1(ˆ
ˆˆ
nnpp
ppz
21
2211 ˆˆˆnnpnpnp
23
Hypothesis Test for the Difference between Two Population Proportions
Step 3: Find p value1. Must be two independent random samples;
both are large samples:
And
2. When the above conditions are met, use Z-Table to find p-value.
10)ˆ1(,ˆ 11 pnpn 10)ˆ1(,ˆ 22 pnpn
24
Example: Bike to School
For 80 randomly selected men, 30 regularly bicycled to campus; while for 100 randomly selected women, 20 regularly bicycled to campus.
Find the p-value for testing:Ho: p1 = p2 vs. Ha: p1 > p2
Answer: z=2.60, p=0.00471: men; 2: women