Lecture 7 - Intro to statistical hypothesis testing: Samples and sampling distributions Flashcards
what is the range of the correlation coefficient r?
the correlation coefficient r ranges from -1 to 1.
A perfect positive correlation is r = 1
a perfect negative correlation is r = -1
and no correlation is r = 0
How is the correlation coefficient r calculated?
one way to calculate r involves:
1. Multiplying z-scores for each data point
2. summing those products
3. dividing by the number of scores
what factors can affect the strength of a linear correlation?
Factors that can affect the strength of a linear correlation include outliers, the shape of the relationship (linear or non-linear), and restriction of range of either variable. Low reliability of the variables can also lower the observable strength of the correlation.
what is the coefficient of determination?
the coefficient of determination is the square of the correlation coefficient (r ^2) and represents the proportion of variance in one variable that is predictable from the other variable
If four unrelated predictor variables are all correlated with the same criterion variable, how large would the r-values need to be in theory to fully account for variance in the criterion?
The r-values would need to be r = 0.5 because r^2 = 0.25, and four predictors each correlated at r = 0.5 would sum to 1 (100% of the variance)
what are sampling error and sampling variability?
- sampling error: The likelihood that the statistics of a given sample will differ from the population parameters
- sampling variability: the likelihood that statistics of any two samples will differ from one another
what is a sampling distribution of the mean?
a sampling distribution of the mean is a (usually imaginary) distribution that we would get if we took an infinite number of samples and plotted their means in a histogram. It describes the likelihood of different possible results (such as means) and is based on the population’s mean and standard deviation, the number of samples taken, and the size of each sample
What are the characteristics of the sampling distribution of the mean?
the sampling distribution of the mean:
- will usually be normal if the sample size is reasonably large
- has the same mean as the population
- has a standard deviation (standard error) that depends on the population standard deviation and the sample size
How is the standard error of the mean calculated?
the standard error of the mean is calculated using the formula (see photo)
- where σ is the population standard deviation and N is the sample size
Given a population distribution with a mean (μ) of 50 and a standard deviation (σ) of 10, what will the sampling distribution be like for a sample size of 25?
shape: normal
Mean (μM): Same as the population mean (50)
standard error (SE): see photo attached
if a variable is normally distributed in the population with a mean of 70 and a standard deviation of 20, what is the probability of obtaining a sample mean of 80 or higher from a sample of 25 people?
- Population mean (μ): 70
- Population standard deviation (σ): 20
- Standard error (SE): see photo
- convert the sample mean to a z-score: see photo
using a z-table, the area above a z-score of 2.5 is 0.62%. Thus, the probability of obtaining a sample mean of 80 or higher is 0.62%
within what limits would the central 95% of sample means fall for the same distribution and sample size?
- the central 95% of a normal distribution corresponds to z-scores of -1.96 and +1.96
- convert these z-scores to raw scores:
- lower limit = 70 + (-1.96 x 4) = 62.16
- upper limit = 70 + (1.96 x 4) = 77.84
Therefore 95% of sample means would fall between 62.16 and 77.84
