Psych 6 Chapter 6
Equation for total number of samples possible in a population from experimental sampling
# of samples possible = N! ÷ n!(N-n)!
Equation for total number of samples possible in a population from theoretical sampling
# of samples possible = N^n n = sample size N = population size
Standard Error (of the mean) how to compute?
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 to compute the SE, take the square root of the variance √σM
Sampling Error
the extent to which sample means selected from the population differ from one another Measured by the STANDARD ERROR of the mean
by graphing the number of the sample mean on the x axis and the frequency of times it occurs on the y axis, what happens?
they form a normal distribution regardless of the distribution of scores in the population
equation for the z score for a sampling distribution
z = (M-µM) ÷ σM M = sample mean µ = mean of the population which is also the mean of the sampling distribution σM = standard error (calculated by σ/ √n)
Minimum Variance (variance for a sampling distribution)
σM² = (M-µM)² / Nⁿ M is the sample mean in each possible sample µM is the mean of the sampling distribution (will be equal to the population mean) Nⁿ is the total number of possible samples that can be selected N = population size n = sample size
Equation to calculate the variance of the sampling distribution of sample means
σM² = σ² ÷ n σM = mean of the sampling distribution (will be equal to the population mean) σ² = population variance n = sample size
3 reasons why sample mean makes a good estimate of the value of the population mean
1. Unbiased Estimator-- On average, the sample mean will equal the value of the population mean 2. 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 is normally distributed 3. A distribution of sample means has minimum variance. The sampling distribution of sample means will vary minimally from the value of the population mean.
Sampling with Replacement More or less random than sampling without 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 A more random sampling method than Sampling without replacement, but it doesn't typically matter because populations used for sampling are typically huge enough that the difference is barely noticeable.
Sampling without Replacement
After selecting a square, do not replace the square before selecting a second square Unbiased sampling method The probability of each selection is conditional. The probabilities of each selection are not the same 1/8 --> 1/7 ----> 1/6
______ ______ _______ is a more unbiased random sampling method than ______ ______ _______
sampling without replacement is a more random sampling method than Sampling with replacement, but it doesn't typically matter because populations used for sampling are typically huge enough that the difference is barely noticeable.
Theoretical Sampling
Order matters and they do replace each selection before the next draw
How to avoid bias?
Use a random procedure to select a sample from a given population All participants must have an equal chance of being selected
a sampling distribution for the mean/variances of a population is ?
a distribution of all sample means/variances that could be obtained in samples of a given size from the same population
Unbiased estimator
any sample statistic obtained from a randomly selected sample that equals the value of its respective population parameter on average. The sample mean is an unbiased estimator because it equals the population mean on average M = ∑x ÷ n, M = µ on average
The Law of Large Number
explains that the larger the sample size, the smaller the standard error Larger samples are associated with closer estimates of the population mean on average
what does z score of a sampling distribution tell us
it is used to determine the likelihood of measuring a particular sample mean, from a population with a given mean and variance.
the larger the standard deviation in the population, the _________ the standard error
larger
Experimental Sampling
order doesn't matter and they do not replace each selection before the next draw
On average, we can expect the sample mean from a randomly selected sample to be equal to the _______ ______
population mean
Central Limit theorem and what it implies
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 this theorem is nice because it let's us know that at least 95% of all possible sample means we could select from a population are within two standard deviations of the population mean... BIG IMPLICATION
Sample Design Two questions associated with Sample design
specific plan or protocol for how individuals will be selected or sampled from a population of interest Asks questions: 1. Does order of selection matter? 2. Do we replace each selection before the next draw?