1. Is it statistically significant?
Hypothesis testing
Is it statistically significant?
Most of the slides are adopted from:

2. Example In a trial a jury must decide between two hypotheses.
Null hypothesis H0: The defendant is innocent
Alternative hypothesis H1: The defendant is guilty
The jury must make a decision on the basis of evidence presented.
In the language of statistics convicting the defendant is called rejecting the null hypothesis in favor of the alternative hypothesis. That is, the jury is saying that there is enough evidence to conclude that the defendant is guilty.
If the jury acquits, then it is stating that there is not enough evidence to support the alternative hypothesis.

3. Goal
Assess the evidence provided by data in favor of some claim (null hypothesis) about the population.
The procedure begins with the assumption that the null hypothesis is true.
The goal is to determine whether there is enough evidence to infer that the alternative hypothesis is true, or the null is not likely to be true.
Results of the test are expressed in terms of a probability
This probability measures how well the data and the hypothesis agree.

4. Example usage in our work
Usually we need to find if the performance of our proposed algorithm is statistically superior than that of a base line algorithm.
General procedure
For i=1:n
Generate a random train/test dataset.
Run both algorithms and get performance numbers.
Record the difference between the performance of the proposed and base line algorithms. Let it be x.
Null hypothesis H0 : x <= 0 i.e. base line better than proposed algorithm.
Alternate hypothesis H1 : x > 0 Our algorithm is better.
We want to reject H0 hypothesis in favor of H1 with certain confidence value.

5. Statistical Example
Consider mean demand for computers during assembly lead time. The operations manager wants to know whether the mean is different from 350 units.
Our test statistics is over mean demand.
Null hypothesis: H0 = 350
Our alternative hypothesis becomes: H1 ≠ 350
Assume σ = 75 and the sample size n = 25, and the sample mean is calculated to be
A large value of (say, 600) provides enough evidence.
If is close to 350 (say, 355), then this does not provide a great deal of evidence to infer that the population mean is different than 350.

6. Is the Sample Mean in the Guts of the Sampling Distribution?
Statistical Example
Assume H0 : μ = 350 is true.
Determine the sampling distribution of sample mean .
Assume has mean
Variance of is /sqrt(25) = 15
Is the Sample Mean in the Guts of the Sampling Distribution?

7. Hypothesis Test 1: Unstandardized test
For the guts to be the center 95% of the distribution [α = 0.05].
The critical values that define the guts will be 1.96 standard deviations of X-Bar
UCV = *15
= 379.4
LCV = 350 – 1.96*15
= 320.6

8. Hypothesis Test 2: Z-score
Calculate z score of the sample mean.
Z = ( )/ = ( – 350)/15 = 1.344
Is this Z-Score in the guts of the sampling distribution?

9. Hypothesis Test 3: p-value
Increase “Rejection Region” until it “captures” sample mean.
P( > ) = P(Z > 1.344) =
p-value = double of this area for two tailed test =
Since rejection region is defined to be 5% and our sample mean is in the 18.02% region, it in not in rejection region.

10. Statistical Conclusions
Unstandardized Test Statistic
Since LCV (320.6) < (370.16) < UCV (379.4), we reject the null hypothesis at a 5% level of significance.
Standardized Test Statistic
Since -Z/2(-1.96) < Z(1.344) < Z/2 (1.96), we fail to reject the null hypothesis at a 5% level of significance.
P-value
Since p-value (0.1802) > 0.05 [], we fail to reject the null hypothesis at a 5% level of significance

11. Interpreting the p-value
Overwhelming Evidence
(Highly Significant)
Strong Evidence
(Significant)
Weak Evidence
(Not Significant)
No Evidence
(Not Significant)

12. Assumptions
Distribution of test statistics follows standard deviation.
Generally one appeals to the central limit theorem to justify assuming that a test statistic varies normally.
Central limit theorem (CLT): mean (x bar) of a sufficiently large number of independent random variables (x), each with finite mean and variance, will be approximately normally distributed.
If the variation of the test statistic is strongly non-normal, a Z-test should not be used.
should be known or can be estimated reliably.
when is same for each x and all x are independent from each other. [Follows from var(c.x) = c2 var(x) and var (x1 + x2) = var (x1)+ var (x2) if x1 and x2 are independent].
A Z-test is appropriate when you are handling moderate to large samples (n > 30).

13. Hypothesis Test 4: Student’s t-test
The test statistic (x bar) follows a Student’s T-distribution if the null hypothesis is true.
T-test is appropriate for small sample size (n < 30).
Great if the populations’ standard deviation is unknown.
s= sample standard deviation.
Once a t value is determined, a p-value can be found using a table of values from Student's t-distribution
The Student t-distribution is symmetric and bell-shaped, like the normal distribution, but has heavier tails

14. Student’s t-test
Two sample t-test can be used to compare means of two independent and identically distributed samples.

15. Hypothesis Test 5: Wilcoxon Signed Test
A non-parametric statistical hypothesis test.
Alternative to the paired Student's t-test when the population cannot be assumed to be normally distributed.
H0 : μ = 0
Rank xis based on their magnitude. Let their rank be Ri.
Wilcoxon signed rank statistics φi = I(xi > 0)
Score
Find the critical value for the given sample size n and the wanted confidence level.
Compare S to the critical value, and reject H0 if S is less than or is equal to the critical value.

16. Thank you 

17. Two types of Errors Type I error Type II error H0 T F I II
When we reject a true null hypothesis. That is, jury convicts an innocent person.
P(Type 1 error) = α [usually 0.05 or 0.01]
Type II error
When we don’t reject a false null hypothesis. That is, when a guilty defendant is acquitted.
P(Type 2 error) = β H0 T F
Reject I II
The two probabilities are inversely related. Decreasing one increases the other, for a fixed sample size.