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
How can two groups be formed. Some properties of the groups.
1.) Mutually-exclusive groups / Independent
- Each score in one group is independent of all scores in the other group
- Participants can only belong to one group
- Size of group not necessarily same
2.) Mutually-paired groups / Dependent
- Each score in one group is linked to a score in the other group by either (a) Measured twice in different time (b) Dependency (twins, husbands,etc)
- Size of groups must be the same
Are groups categorical / continous
Categorical
How do we initially examine distribution of scores in both groups
Boxplots:
- Group medians
- Outliers
- Normality
- Homogeneity of Variance
(NO MEAN)
After a boxplot to examine distribution of scores, what is nice to use to check out outliers
qqPlot. Check out spread and outliers.
Since there will always be a difference between sample means of 2 groups, what are we actually finding out
Sample means will ALWAYS be different, but we are actually find out if this
Difference is:
- Due to random sampling variability when groups from same population
- Due to random sampling variability + difference in population means when groups from different populations
Sampling distribution of Group Mean Differences: What do we try to create
Confidence Intervals
Given a sample distribution of group mean difference where M (μ1 - μ1) = 0, what does the confidence interval tell us
The range of plausible values if there is no mean difference
- ( 0 - crit.t*se, 0 + crit. t*se)
Given a sample distribution of group mean difference where M (μ1 - μ1) = 1.25, what does the confidence interval tell us
The range of plausible values given the mean difference of 1.25
- ( 1.25 - crit.t*se, 1.25 + crit. t*se)
- Notice how the values are relative to 1.25
When investigating differences between 2 independent groups, what are the 3 assumptions
- Observations are independent
- Observed scores on the construct measure are normally distributed
- Homogeneity of variance in 2 groups
- Group 1 compared to Group 2
(No assumption of linearity)
