week 8: the distribution of means Flashcards
Distribution of means example
you approach a random sample of 100 locals and assess their IQ and find M = 90
So, the question becomes, what is the probability of finding a sample N=100 with M=90 from a population where μ = 100 and population SD = 15
can we still use z formula?
no because we are now comparing a sample to a population and not a single score to population
Sampling variability
no 2 samples will have exactly the same M or SD
, we really want to minimise sampling variability as much as possible
Minimising error
the easiest way to minimise this error is to use larger sample sizes
(the more data you collect, the better representation you will have of the population of interest)
Minimising error
the easiest way to minimise this error is to use larger sample sizes
(the more data you collect, the better representation you will have of the population of interest)
distribution of means steps
Step 1: Decide on a research (effect) and null (no effect) hypothesis
Step 2: Identifying the comparison distribution
Step 2: Identifying the comparison distribution
Our comparison distribution becomes a distribution of sample means rather than single scores
Why does this distribution normalise
By taking larger samples (increase N), we better approximate the population, providing a better estimate of its mean (μ).
Characteristics of the distribution of means: #1
the mean of the distribution of means is the same as the population mean from which the sample is taken
μm= μ
μm(sample mean)
μ=population mean
the greater the number of sample means that are drawn, the closer the mean of sample means is to the population mean
Characteristics of the distribution of means: #2
the variability of the distribution of the means is less than the variability of the population from which the sample is taken
om
Measuring variability in sample means
where o2m is the variance of the distribution of means
2 is the variance of the population
N is the number of scores in each sample
So can see as N increases, the variance decreases
Standard Error of the Mean
The standard deviation of the distribution of the means
It shows the average amount we can expect our sample mean to vary around the true population mean
Increase N, decrease error example saved to desktop
Characteristics of the distribution of means: #3
the shape of the distribution of the means tends to be unimodal and symmetrical
The normal distribution allows us to place our sample mean on the distribution of sample means using standard scores.
3 types of distributions
population
sample
distribution of means
Hypothesis testing against a known population
last lecture we simply found the Z score of our observation relative to the population distribution
with a sample, we find the Z score of the sample mean relative to the distribution of the means
Estimation and confidence intervals
we can also estimate a range of possible means (around the sample mean) that are likely to include the population mean
the wider the range, the more confident we can be that it will include the true population mean
we want a range that is wide enough to include the population mean, but NOT so wide as to be meaningless
this range of possible means = confidence interval or CI
e.g., to be 95% sure, you want a 95% confidence interval
note: the upper and lower ends of the confidence interval are called the confidence limits
to calculate confidence intervals (CI)
- determine the characteristics of the distribution of means (estimated mean and variance)
- use the normal curve table to find the Z scores to match the upper and lower percentage that you require (95% CI = +/-1.96)
- convert the Z scores to raw scores on your distribution of means (to obtain upper and lower confidence limits)
