2-tailed test hypothesis test
H0: π = π
Ha: π β π
1-tailed test hypothesis test
H0: π = π
Ha: π > π OR Ha: π < π
Z-score
measures how many standard deviations away from the mean your measurement is
π§ = π β π / π
Significance level (πΌ)
percentage of the data you want to be above or below you measurement in order for it to be significantly different from the mean
(most often: πΌ = 5%)
Confidence level equation
1 - πΌ
p-value
the area of the tail beyond your measurement
- the probability of obtaining the value as extreme or more extreme as the given value, provided that H0 is true
How to Calculate the p-value?
1) Z-table or T-table
2) Excel using 1- normal or 1- standard normal dist
π β€ πΌ
reject H0, your value is significantly different from the mean
π > πΌ
fail to reject H0
Sample mean is a random variable with
standard deviation, this is called standard error
Assumptions of Z-test
(What if not all of these assumptions are met?)
- Sample data follow a normal distribution
- All observations are independent
- Sample size is more than 30
- Population standard deviation is known
If these are not met then use a t-test
Types of two samples
- well-separated distributions
- overlapping distributions
- different variances
The confidence interval of two samples
π Β± π‘-critical * π /βπ
Hypothesis Tests for Variances
- Hypothesis test for single variance β use π2 distribution
- Hypothesis test for equality of two variances β use F-distribution
ππ (chi-squared) distribution
distribution of the sum of π independent standard normal random variables
F-distribution
distribution of a ratio of two random variables which are distributed according to π2-distributions with parameters π1 and π2.
The denominator for the t-statistic depends on whether
the variances of the samples can be assumed to be the same so we need a separate test for variances
- There is a equation for equal and unequal t-statistic
F-test
can be used to test if two variances are equal or if one variance if greater than the other
(left tail if you divide smaller by larger variance, right tail if you divide larger by smaller variance)
π2 test
can be used to test if the variance of a population is equal to a specified value, based on the sample data
