PSYC*1010 Chapter 7: Probability and Samples Flashcards
What is a sampling error?
The natural discrepancy or expected amount of error between a sample statistic and its corresponding population parameter
What is a distribution of sample means/sampling distribution?
The collection of sample means for all possible random samples of a particular size (n) that can be obtained from a population
What are the three steps involved in constructing a distribution of sample means?
- Select a random sample of a specific size from a population
- Calculate the mean for that sample
- Continue selecting samples and calculating means until all possible random samples have been included
T or F: Sample means in a distribution should be less frequent around the population mean.
False. They should be most frequent around the population mean
What type of distribution do sample means form?
A normal distribution
As sample size increases, how do sample means change in relation to the population mean?
The larger the sample size, the closer the sample means should be to the population mean
T or F: Means obtained from small samples should be more widely scattered in a distribution.
True
What is the central limit theorem?
A mathematical theorem that specifies the characteristics of the distribution of sample means
According to the central limit theorem, how will the mean of a distribution of sample means compare to the mean of the population?
The distribution of sample means will have the same mean as the population (μ)
According to the central limit theorem, how will the standard deviation of sample means compare to the standard deviation of the population?
The standard deviation of sample means is equal to the population standard deviation divided by the square root of the sample size (σ/√n)
What does the central limit theorem state about approaching a normal distribution of sample means?
Sample means will approach a normal distribution as n (sample size) approaches infinity
The value of the central limit theorem comes from what two simple facts?
- The theorem describes the distribution of sample means for any population regardless of shape, mean, or standard deviation
- The distribution of sample means approaches a normal distribution very rapidly
At what point/sample size is the distribution of sample means almost perfectly normal?
When n=30
One of which two conditions must be met to assume normal distribution of sample means?
- The population from which the samples were selected is normally distributed
- The number of scores in each sample is 30 or more
What is the expected value of M?
The mean of all sample means
Why is the expected value of M an example of an unbiased statistic?
Because on average, the sample statistic produces a value exactly equal to the corresponding population parameter
What is the notation for mean of sample means/expected value of M?
μ with subscript M
Why is the mean of the distribution of sample means typically denoted as μ, even though it has its own notation?
Because the distribution of sample means is always equal to μ (population mean)
What is the standard error of M?
The standard deviation of the distribution of sample means
What is the notation for standard error of M?
σ with subscript M
What does the standard error of M provide a measure of?
How much distance is expected, on average, between the sample mean and the population mean
What does a small standard error indicate?
All sample means are close together and have similar values
What does a large standard error indicate?
Sample means are scattered over a wide range and there are big differences from one sample to another
Can standard error measure how well an individual sample mean represents the entire distribution?
Yes
What two factors determine the magnitude of the standard error?
- The size of the sample
- The standard deviation of the population from which the sample was selected
In relation to standard error, what does the law of large numbers state?
The larger the sample size (n), the more probable it is that the sample mean will be close to the population mean
Does standard error increase or decrease in relation to the sample size?
Standard error decreases as sample size increases
T or F: Standard error can be reduced by increasing sample size to around n=30, but increasing size above that doesn’t provide much additional improvement.
True
What value for n produces the smallest possible sample from a population and largest standard error?
n=1
As sample size increases, what happens to standard error?
It decreases
What is the equation for standard error?
σM= σ/√n
What is the primary use of the distribution of sample means?
To find the probability associated with any specific sample
What does the z-score for sample means describe?
The distance of a sample mean from the population mean in terms of the number of standard deviations
What is the equation for calculating the z-score of a sample mean?
z= (M- μ)/σM
How does the z-score formula for a score in a population differ from the z-score formula for a sample mean?
Standard error must be used in the denominator when calculating z-scores for sample means, but standard deviation is used for a single score
When the distribution of sample means is normal, what can be used to determine the probability of a sample mean occurring?
The z-score and unit normal table
