Discrete Uniform
'N' Possible values, each with equal probability
Continuous Random Variable
Can be any real number
Law of small numbers
Number of observations 'N' is large and probability 'P' is small.
Distribution is dependent on the mean
λ=N×P
Poisson distribution with parameter λ
Law of large numbers
When the number of events 'N' increases, the relative freqency better estimates the theoretical probability of an event
Pearson's Chi-square test
Test of independence
Test of Homogeny
Cannot be one-sided
Fisher's exact test
Test of Independence
Test of Homogeny
Can be One or Two-sided
Binomial
Used when sample size and prevalence are given
Popper's theory of falcification
A theory can be considered scientific, only if it can be experimentally disproved
Wilcoxans rank sum test
Testing the orders (ranks) of the observed values
T-test
Difference of the averages
Assume normal distribution
Standard error of Mean =
σ/√n
σ = Standard Deviation
n = Number of Values
Interval Estimate =
Confidence interval
Contains the unknown quantity with high probability (95%)
Two sample t-test
Investigates a hypothesis about a relationship between two populations with two samples independent from each population
Paired sample t-test
Same as two-sample t test only with paired samples between the groups
Difference between the pairs must be normally distributed
Levene Test
Variances of several populations
Independent Samples
H0 claims each population variance is the same
Mood's Median
Median of several populations
H0 can be one or two sided in case of two samples
Must be continuous
Which hypothesis can never be rejected?
The Alternative Hypothesis
What can be the alternative hypothesis for a correlation test?
Correlation doesn't = 0
The question 'Is there a difference between...' leads to a...
Two-sided test
ANCOVA
Corresponds ot the notation:
X5, X4, X3, X2, X1 → Y
Two types of conditional probability
Sensivitiy - % Sick individuals diagnosed
Specificity - % Healthy individuals diagnosed
Bocyes Theorem
Calculation method for inverse conditional probabilities
Point Estimate
Single number calculated from the sample
Ordinal Scale
When the data being collected is qualitative but can be ranked
Ratio Scale
Used when the data is relative to a predetermined value such as length or mass
Interval Scale
Allows calculation of the degree of difference between items, but not the ratio
e.g degrees Celcius
ANOVA
Used to investigate a hypothesis with the mean values of several populations
Measures equality of means in 4 groups
H0 claims each population mean is the same
F-test
Investigates the hypothesis about the relationship of two population variances based on two independent samples
Used for multiple sample problems
H0 claims each population variance as the same
Compare two side samples and one side samples
Two sided samples are symmetrical
One sided have a definitive direction
Type 1 error
Rejection of a true null hypothesis
False-Positive finding
Type 2 error
Retaining a false null hypothesis
False-negative finding
Power of the statistical test
Gives probability of rejecting the null hypothesis where the alternative hypothesis is true
Chi-Square test
Used for Large Samples
Used to compare expected and observed values
E.g Flip a coin 100×
Expected values - H:50 T:50
Observed Values - H:54 T:46
Give the formula for the Chi-square test
A & B are independent if...
P(A)P(B) = P(A∩B)
Their joint probability equals the product of their probababilities
Give the formula for conditional probability
P(A) = P(A|B)
Explain the addition rule
A die is rolled, what is the probability of getting a 2 or a 5?
- P(A or B) = P(A) + P(B)
- P(A) = 1/6......P(B) = 1/6
- P(A or B) = 1/3
Nominal Scale
When data is collected qualitatively
e.g Eye colour, nationality
What is the approximate probability that the value of a normally distributed random vatiavle is vetween (mean-2 standard deviation) and (mean+2 standard deviation)?
95%
The relationship P(A+B)=P(A)+P(B) is true if...
A and B are mutually exclusive
The relationship P(A+B)=P(A)+P(B)-P(A×B) is true if...
A and B are not mutually exclusive
Normal distribution is...
A function that represents a distribution of many random variables as a symmetrical bell shaped graph
Standard deviation is...
A quantity calculated to indicate the deviation of a group as a whole
Give the formula for standard deviation
Variance is...
Standard Deviation squared
Which scale is typical for a correlation coefficient?
-1 < X < 1
Which test should be used if the equality of means in four groups (many) is to be analysed?
ANOVA
Which test is appropriate for only large samples?
Chi-square test
Which part of statistics does the chi-square test belong to?
Descriptive Statistics
Inductive statistics
Dealing with generalisations, predictions, estimations and decisions from the initial data
Descriptive Statistics
The use of clear and concise numbers to describe a set of known data
Describe odds ratio
The ratio of odds between two groups
e.g. Odds of colourblindness in women 4:48
Odds of colourblindness in men 1:47
(4/48)/(1/47) = 3.91 (Men 4x more likely than women)
Explain relative frequency
If a team plays 12 games of football and wins 9, relative frequency of winning is 3/4
Formula to calculate relative frequency
RF = K/N
K = Number of observed events
N = Number of trials
Conditions of applicability of ANOVA
Normally distributed variables for each group, Variances should be equal in each group
What is the critical value?
The value obtained if chance alone contributed to the outcome
What is the Observed Value?
The value collected experimentally
What is the p value?
Probability of finding the observed result when the null hypothesis is true
