Chapter 6- Probability and the Central Limit Theorem Flashcards
sampling distribution
a distribution of all sample means that could be obtained in samples of a given size from the same population.
- Sampling distributions are made by repeatedly plotting sample statistics.
Sampling without replacement
a method of sampling in which each participant or item selected is not replaced before the next selection. This method of sampling is the most common method used in behavioral research.
Sampling with replacement
a method of sampling in which each participant or item selected is replaced before the next selection. Replacing before the next selection ensures that the probability for each selection is the same. This method of sampling is used in the development of statistical theory.
sample design
a specific plan or protocol for how individuals will be selected or sampled from a population of interest.
central limit theorem
explains that regardless of the distribution of scores in a population, the sampling distribution of sample means selected at random from that population will approach the shape of a normal distribution, as the number of samples in the sampling distribution increases.
standard error of the mean, or standard error
he standard deviation of a sampling distribution of sample means. It is the standard error or distance that sample mean values deviate from the value of the population mean.
Sampling error
the extent to which sample means selected from the same population differ from one another. This difference, which occurs by chance, is measured by the standard error of the mean.
law of large numbers
states that increasing the number of observations or samples in a study will decrease the standard error. Hence, larger samples are associated with closer estimates of the population mean on average.
The sample mean has the following three characteristics:
The sample mean is an unbiased estimator. On average, the sample mean we obtain in a randomly selected sample will equal the value of the population mean.
A distribution of sample means follows the central limit theorem. Regardless of the shape of the distribution in a population, the distribution of sample means selected at random from the population will approach the shape of a normal distribution, as the number of samples in the sampling distribution increases.
A distribution of sample means has minimum variance. The sampling distribution of the mean will vary minimally from the value of the population mean.
The variance of the sampling distribution of sample means
equals the population variance divided by the sample size
σ ²/n… The σ is squared because its the variance, so need to undo the square root calculation that was done when finding the standard deviation.
Explain the relationship between standard error, standard deviation, and sample size
As the population standard deviation (σ) increases, standard error increases. Hence, the farther scores in a population deviate from the mean in a population, the farther possible sample means can deviate from the value of the population mean.
As the sample size (n) increases, standard error decreases. Hence, the more data you collect, the closer your estimate of the value of the population mean. This relationship is explained by the law of large numbers.
Calculate the standard error of the mean
Conditional Probability: The “AND” RULE
P(A ∩ B) = P(A) × P(B)
When we want to find the probability of A and B, we multiply the individual probabilities together.
This will make the probabilities smaller; rarer.
Conditional Probability: The “OR” Rule
P(A ∪ B) = P(A) + P(B)
When we want to find the probability of either A or B, we add the individual probabilities together.
the Gamblers Fallacy
The belief that something is less likely to happen if it follows a series of similar events or that past events change the probability of future ones.
These events are all independent of each other, meaning that the probability does not go down because one event has already occurred several times.
