W6: Independent and Dependent t-tests Flashcards
What are the 4 most common continuous probability distributions
- Normal
- Chi-Squared
- Positive Skew
- (Student’s) t
- Assymmetrical Skew
- F
- Positive Skew
What kind of probabiity distributions are there? What are they defined by?
- Continuous / Discrete
- Univariate / Multivariate
- Univariate = 1 particular statistic
- e.g. variance
- Multivraite = >1 statistics
- e.g. covariance
- Univariate = 1 particular statistic
- Central / Non-Central
- Central
- Under h0
- Non-Central
- Under H1
- Central
They are defined by their repsective parameters
What are 2 ways of expressing probability distributions. How do they relate to each other
- Density Functions
- E.g. Bell-shaped Curve
- Cumulative Distribution Functions
- p values
- For continous distributions, Cumulative Distribution Function is obtained by integrating Density Functions
What is the normal density function defined by (in terms of parameters)
- ) Mean (μ)
- ) Standard Deviation (σ)
What is the difference between a “normal distribution” and a “standard normal distribution”
Standard Normal Distribution
- Mean (μ) = 0
- Standard Deviation (σ) = 1
How do change a normal distribution into a standardised normal distribution. What is it called
Argument Z or Z statistic
- Standardard Normal Variate
- It is transforming the x argument into a standardized Z value
- Z = (x - μ) / σ
- Note: Capital Letter Z
- While the formula is the sam as a z-score, it is NOT a z-score, but a z-distribution
- Mean: 0; SD: 1
How do we calculate probabilities in a density function
Area:
- Below any value on x-axis
- Above any value on x-axis
- Between any 2 values on X-axis
What is so important about a probability distribution. Think through… The answer is long (Mathematically and Practically)
Mathematically
- A sampling distribution of a sample statistic will correspond to a particular probability distribution
- i.e., 1-1 relationship between sampling distribution and probability distribution
Practically
- We do not have to construct a sampling distribution to undertake research. Since we only have one sample and one summary characteristic, we can use probability distributions as a basis to knowing what the distribution of a sample statistic (sampling distribution) will be, and using that probability distribution, we can
- Construct CI
- Calculate P values
What are probability distributions used for
- ) Constructing confidence intervals
- ) Calculating P values for null hypothesis tests
Correlation coefficients, Cramer’s V, Odds ratio, Regression Coefficients, R2, mean difference
What are they distributed as
t (df)
- Correlation coefficients
- Regression coefficients
- Mean differences
Chi-Squared
- Cramer’s V
N (logOR, σ)
- Odds Ratios
F (df1, df2)
- R2
How do we calculate confidence intervals from probability distributions. What do we need?
Example: 95% CI in t-Statistic
- Standard error.estimate
- Derived from sample statistic (b^ )
- σ^b
- Critical t-value
- Derived from desired confidence
- 95% CI = t.975
- (1 + .95)/2
ME
- ME = σ^b x t.975
- (b^ - ME) <= b^ <= (b^ + ME)
What kind of Margin of Error for Confidence Intervals do we normally get when it is derived from t-statistic probability distribution
Symmetrical Margin of Error
(b^ - ME) <= b^ <= (b^ + ME)
- Gurantee that 95% of all CI constructed from repearted samplings from the same population will contain the true population parameter value
When we think of a difference, we are interested in groups differing in _______
When we think of a difference, we are interested in whether groups differ in terms of their respective POPULATION MEANS
When phrasing RQ in terms of a difference, must we state which group is higher or lower
We can, but its not necessary.
We are interested whether there’s a differences
When we think of a difference, we are essentially asking whether members of each group are
- ) Distinct population defined by different means
- ) Subgroups within the same population with the same mean
