Week 2 - Lecture 2 Flashcards
Hypotheses Testing
define conceptual hypothesis
- outlines the predicted relationship between the independent varible (IV)/predictor variable and the dependent variable (DV)*/outcome variable *
- eg. Intoxicated participants will be more creative than sober participants
define statistical hypothesis
- represents the mathematical relationship **presumed to exist **between **two or more population parameters **
- eg. creativity scores will be higher among intoxicated participants than among sober participants
what is the Neyman-Pearson Hypothesis Testing Model?
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statistical analysis in psychology is guided by the Neyman-Pearson hypothesis testing model
In this model there are **two statistical hypotheses offered: ** - The null hypothesis (H0)
- The alternative (research) hypothesis (H1)
what does the null hypothesis (H0) represent?
- there is no relationship between the two variables that we are investigating
- assertion that researcher is trying to reject (something researchers want to say is unlikely to be true)
what does the alternative (research) hypothesis (H1) represent?
- there is a relationship between the** two variables**
- what the researcher believes to be true
how does the null hypothesis vs alternative hypothesis look graphically
- yellow could be drawn anywhere
- statistically you’re testing the** likelihood** that the sample came from the population as specified in the null
What is meant by “rejecting the null”?
the question we are always considering is:
- what is the likelihood my **sample was drawn **from a population with the mean specified under the null hypothesis?
- this is why we must **specify **the **H0 mean **up front
- we then aim to reject it on the basis that the sample mean is so extreme that it is unlikely to have come from the population **specified under the null **
How “extreme” does a sample mean need to be for us to “reject the null”?
By convention, we** reject the null** when the sample mean is so extreme that it would occur when the** null is true** less than 5% of the** time**
- Referred to as** p < .05** (critical region, alpha greek symbol)
- specifies acceptable risk of **rejecting H0 when it is true**
- ie. only 5 out of 100 experiments would have produced the obtained result (or an even more extreme one) by chance (under H0)
what does the critical region look like mapped into the normal standard distribution curve?
when you get a Z score in the middle region, we retain H0
- It’s not extreme enough to feel confident that it’s unlikely the sample came from the population *
Once it gets into the rejection region of the two tails, it’s unlikely *we would** get a value this extreme** based on chance alone, if H0 was correct
- spread our error into two tails, so that** 2.5% of the time we’re going to make an error in either direction **
what is the significance/meaning behind rejecting/retaining the null hypothesis (H0)?
- rejecting H0 simply means that it is unlikely given the current data and we accept the alternative hypothesis for the time being
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retaining H0 does not mean that it is true - we might just not have found sufficient evidence to reject it at the moment
**We cannot prove a statement ** - the best we can do is evaluate the evidence to see whether H0 is unlikely to be true
Four types of hypotheses
define causal-directional hypotheses
- more A causes more (or less) B
- eg. consuming alchohol increases creativity
- experimental data
Four types of hypotheses
define causal non-directional hypotheses
A causes B
- eg. consuming alchohol affects creativity
- experimental data
Four types of hypothses
Define non-causal directional hypotheses
- more A is related to more (or less) B
- Greater alchohol consumption is associated with greater creativity
Four types of hypotheses
Define non-causal non-directional hypotheses
A is related to B
- alcohol consumption is associated with creativity
which type of hypothesis is the following scenario?
- Giving a gift to someone impacts your happiness
Causal, non-directional
which type of hypothesis is the following scenario?
- setting specific, measurable goals leads to better performance
causal, directional
which type of hypothesis is the following scenario?
- stress is related to workplace attitudes
non-causal, non-directional
what is a non-directional hypotheses?
Predict that a **difference will exist without specifying the nature or direction of that difference**
- eg. alchohol consumption is associated with creativity
- eg. relatable examples affect enjoyment learning statistics
However, we’re always interested in the direction when it comes to **interpreting any significant effects **
define one-tailed vs two-tailed tests
When Alpha is 0.05, acceptable chance of making a** Type 1 error **(rejecting H0 when H0 is actually true)
- One-tailed test: Large rejection region just on one side
- Two-tailed test: Smaller rejection regions on both sides
- **allows for differences in either direction (more room for error) **
why do we compare a** sample mean** to a distrubution of means in psychology?
Discussed** individual scores** of their l**ikelihood of being observed **in the **general population **
- compared one individual score (yours) to a distribution of individual scores (the population)
- But in psychology we’re interested in sample scores, *not *a single *individual’s score *
Statistically, we can’t compare a sample mean with a comparision distribution of individual scores
- instead we use a comparison distribution of means of samples of scores: we compare a sample mean to a *distribution of means *
what are sampling distributions?
- sampling distributions are how things would look if you repeatedly took samples (of a given size) from a **population of interest **
- It’s not representing the raw scores but some **statistic from the sample **
- Can have a sampling distribution for any statistic, but the most common is the** sampling distribution of the mean **
what is natural variability?
Gives you the **distribution of means **we can expect when we do this repeated process of drawing samples of the same size and calculating their mean
- mean of the sampling distribution will be the population mean, but the SD shows you there is **variability around that **
When will the sampling distribution of the mean be normally distributed?
- the sampling distribution of the mean will be normally distributed if the population of raw scores is normally distributed or if the** sample size is at least 30 (central limit theorem)**
- don’t need to understand central limit theorem, just know that sample size matters
- **bigger sample size **= sampling distribution of the mean will be normal
- *critical assumption for our statistical tests *
- the mean of the sampling distribution is equal to the mean of the actual population (refer to photo)
what is the standard error of the mean?
- Standard error of the mean is the** standard deviation **of the **distribution of sample means **
- It represents the typical or average distance between a **sample mean **and the mean of the population
- It is used to* define *and accurately measure sampling error
- Tells you how much you would expect the sample mean to vary just **based on chance alone **
define standard error (SE)
how much would you expect the sample mean to differ from the population mean, just based on chance alone, for a sample of this size
True or False
We have a standard error of 3. This means
we would expect the **mean IQ **of our sample to differ from the population mean on average by 3 IQ points. So a sample mean IQ of 97 (3 points below the mean) or 103 (3 points above the mean) is within the margin of error we would expect based on sampling error
True
what does the Z score tell you?
tells you in SE units how far the sample mean is from the population mean
shortcut for rejecting the null
critical z score (when doing a two-tailed test, and alpa = 0.05):
- 1.96 is the critical value on either side you need to exceed to reject the **null **hypothesis
- any z-score falling above 1.96 = you can **reject the null **
- any z-score falling **below -1.96 **= you can reject the null
- any **other z-score **= you need to retain the null
When we know the mean of a population but do not know the standard deviation for the population the best we can do is…
- Use an approximation of that standard deviation from the standard deviation for the sample
- This means we end up using a one-sample t-test
steps to testing hypotheses about means - sigma unknown
- conduct a one-sample t-test
- to calculate SE need to substitute an estimate of sigma
what is the formula for sigma unknown hypotheses testing?
refer to photo
When sigma is unknown, the nature of the test changes how?
- then we obtain a t value and compare to a* t distribution *
WHY? - the sampling distribution of the sample variance is** positively skewed**
-** S^2** will underestimate sigma^2 (esp. for small samples) - We’d be more** likely to make a Type 1 error **if we used critical z scores, so we use **critical t instead **
explain Student’s t distribution
- Gosset showed that if s^2 is used in place of sigma^2 it would lead to a particular distribution, called a t-distribution
What are the similarities in both z and t distributions
Both distributions (z & t):
- have a mean of zero
- are symmetrical
- are unimodal
what are the differences between a z and t distribution?
The t distribution:
- has a flatter distribution
- Has a larger standard deviation
- Depends on df, so t-distribution is a family of distributions
To find the correct ‘critical’ t value you need…
- df (degrees of freedom) and alpha
Implications for statistical inference
- larger values of t required to reject null hypotheses - depends on df
- as df approaches infinity, the distribution will approach normality and be equivalent to z
Degrees of freedom - what are they?
- **number of scores that are free to vary **
- df = N = number of parameters estimated
- For a one sample t-test,** df = N - 1 **
- because in a one-sample t-test we use s^2 to estimate alpha^2
EXAMPLE:
- take the numbers 6,8 & 10
- sample mean (x-bar) = 8
- **Now you may change any of these numbers as long as the mean stays as 8 **
- How many numbers can you change?
- only 2 are “free” to vary and keep the mean of 8
- 6,8,10 (sample mean = 8)
- 7,?,13 (sample mean = 8)
- ? has to be 4
- so for a one sample t-test, **df = N - 1 **
