Null hypothesis (Ho)
Alternative hypothesis (Ha)
Claim we weigh evidence Against
Claim we are trying to find evidence For
One sided alternative hypothesis vs two sided alternative hypothesis
State that a parameter is GREATER or LESS than the Null value > or <
State the parameter is DIFFERENT from Null value unequal
Interpreting P-values
The probability of getting evidence for the Ha as strong or stronger than the observed data, Assuming the Ho is true
How to make a conclusion in a significance test
- If the p-value is small you reject the Ho and conclude you have convincing evidence for the alternative
- If p-value is Large you Fail to Reject the Ho and conclude you do not have convincing evidence
Significance level
Alpha level (α)
α - Boundary for deciding what to do with null
NEVER accept the Null
“guilty” or “not guilty
—>
“reject” or “fail to reject”
CAUTION on alpha level
Alpha level should be decided before you collect data
AP Exam Tip
(3 steps for answer)
You must compare p-value to alpha level
You must decide on the null
You must say it is/isn’t convincing evidence for the alternative
Type 1 error
Type 2 error
1: The test rejects the null when is was true “finds convincing evidence for alternative when null was true”
2: The test fails to reject the null when it was false “Does NOT find convincing evidence for alternative when it was true”
Type 1 error probability
Probability of making a type 1 error is equal to your alpha level
Careful considerations based on errors
- Choose an alpha level based upon which error type 1 or type 2 is more serious. For example if you want to avoid a type one error make your alpha level small
- As the prob of a type 1 error DECREASES the prob of a type 2 error INCREASES and vice versa
- Consider possible consequences of each before choosing an alpha level based
Conditions for performing a significance test about a proportion
1- Random - data comes from random sample
2- 10% - N >_ 10n
3- Large counts - nPo >_ 10 and n(1-Po) >_ 10
Performing a significance test about p
Center: Mp^ = Po
Spread: σp^ = square root Po(1-Po) / n
Shape: Large counts condition checks for normality
Standardized test statistic (formula)
Tells us how many SD’s away is our statistic from the Po / Null value
General:
Stat-Null / SD of stat
Specific:
Z= p^ -Po / square root Po(1-Po) / n
Significance tests: 4 step process
1: Define parameter (with let statement) state both hypotheses
2: check conditions, Name of the test 1 prop z-test
3: state test, stat and p-value
4: write conclusion in context
Power
Probability that a test will find convincing evidence for the Ha, when a certain other parameter value is true
Relating power and type 2 error
Power = 1-P(Type 2 error)
Increasing the power of a significance test
- Increase sample size
- Increase alpha level
- The null and alternative parameter values when further apart, power INCREASES
How to make a conclusion in a significance test (3 conditions
- Random - Data comes from Random Sample
- 10% - N>_10n
- Normal - Pop normal, n>_ 30 (CLT), or graph data with no heavy skew or outliers
t statistic
t = x bar - Mo / Sx / square root n
Table B limitations
or tcdf limitations
df=n-1
If df isn’t provided in Table B use next lower df that is available
t distribution
A t-dist is symmetric, single peak, bell-shaped, But it has More Area in its tails
