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What does parsimony dictate?
- We should always choose the simplest possible model
- In psychology, the evidence must establish a need for the full model beyond reasonable doubt
What must be considered before choosing a full model?
- Does the difference between means reflect a genuine signal in the data?
- Is the apparent signal merely a result of sampling error or noise?
How can uncertainty be quantified?
- Uncertainty can never be eliminated, and data will always be noisy
- Probability theory is employed
What is the Quincunx? (bean machine)
- Invented by Frances Galton
- Serves as a manual model of residuals
- Balls are dropped into the top through a funnel, strike one of the many pins and fall into one of the buckets at the bottom
The number of equally probable ways that an event can occur, divided by the total number of equally probable outcomes
How is probability calculated using the multiplication rule?
- Events are assumed to be independent
How is probability calculated using the addition rule?
P(A or B)=P(A)+P(B)
What denotes the number of trials, with regards to binomial distribution?
What does P refer to, with regards to binomial distribution?
- The probability of one of the two possible outcomes occurring
- E.g (Head or tails)
What does r denote, with regards to binomial distribution?
An index of the number of times the specified number of outcomes occurs
Write an equation for the binomial distribution
- This need not be memorised as it is built into minitab
What is the binomial coefficient?
What happens as the number of observations increases?
- The observed proportion provides an increasingly accurate estimate of the probability
- This is known as the law of large numbers
What effects does increasing the sample size have?
The probability that a sample size will be close to the parameter value increases
How can probability be interpreted?
Long run relative frequency or population proportion
What is the formula for population proportion?
What happens as the number of trials increases?
The relative frequency approaches the population (true) value
How can a sample be said to be random?
Each element has an equal chance of being selected
What happens as the rows on a quincox become infinite in number?
- The outcome variable goes from being discrete to being continuous
- The binomial distribution tends towards normality
What claim is made by the central limit theorem?
A variable that is the net result of adding together a large number of contributing independent outcomes will have an approximately normal distribution
Why is the central limit theorem named as such?
- It states what will happen in the limit as the number of outcomes becomes infinite
- It is central as it is fundamental
Other than variables that come about due to addition, which other variables fall under the central limit theorem?
- Attributes that are influenced by a large number of independent factors that add to determine the final measurement
- E.g: Height
- This applies to other psychological measurements too
How are continuous variables different than discrete ones?
They have an unlimited number of possible values
Why is it inappropriate to ask for the probability associated with a single continuous value?
With continuous variables, statements of probabilities are always about ranges of values
For continuous variables, how are probabilities represented?
- Areas under the curve
- This means that the height of the distribution at any point is not the probability, but in fact the probability density