Statistics Concepts 2

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the collection of all people, objects, or events having one or more specified characteristics

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Population

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a proper subset of a population

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Sample

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the probability distribution of a statistic (this is to distinguish it from a probability distribution for, say, a score value)
Any statistic calculated on a sample has a certain sampling distribution

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Sampling distribution

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a descriptive measure for a population

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Parameter

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is a descriptive measure for a sample

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Statistic

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Describes the sampling distribution of the mean
1. μx=μx The mean of the sampling distribution is the same as the mean of the population. The expected value of the sample mean is the same as the population mean.
2. (variance of the mean)

(standard error of the mean)
The variance of the sampling distribution is proportional to variance of the population, and inversely related to sample size.
3. As n↑ the sampling distribution approaches the shape of a normal distribution, regardless of the shape of the population distribution.

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Central Limit Theorem

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the standard deviation of the sampling distribution of a statistic.

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Standard error

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(i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all possible samples (of a given size) drawn from the population. Secondly, the standard error of the mean can refer to an estimate of that standard deviation, computed from the sample of data being analyzed at the time.

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Standard error of the mean

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sampling that is representative, unbiased- how true

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Accuracy

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sampling depends on size of the sample, width of confidence band. Bigger samples yield more precise inferences.

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Precision

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involves the use of sample data to calculate a single value (known as a statistic) which is to serve as a "best guess" or "best estimate" of an unknown (fixed or random) population parameter.

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Point estimate

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is the use of sample data to calculate an interval of possible (or probable) values of an unknown population parameter, in contrast to point estimation, which is a single number.

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Interval estimate

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is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval (i.e. it is calculated from the observations), in principle different from sample to sample, that frequently includes the parameter of interest, if the experiment is repeated.

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Confidence interval

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a prediction that there is no difference between groups in a population, or there is no relationship between variables in the population

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Null hypothesis

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is a prediction that there is a difference between groups in a population, or there is a relationship between variables in a population

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Alternative hypothesis

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rejecting a null hypothesis when it is true (like putting an innocent man in jail) a false positive

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Type I error (α)

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fail to reject a null hypothesis when it is false (like sending a guilty man free)

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Type II error (β)

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- father of modern statistics… Wrote Statistical Power Analysis for the Behavioral Sciences

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Jacob Cohen

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Correct acceptance of a true null hypothesis

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(1-α)

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- the probability of rejecting a false null hypothesis (sensitivity)

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Power (1−β)

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a technique that helps a researcher to determine how big a sample size should be selected. As power increases, the chance of type II errors decreases.
Things that affect Power:
Sample size increases
Alpha increases
Effect size
A priori- prospective power analysis… Estimating sufficient sample size to
achieve adequate power.
Post hoc- retrospective power analysis…

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Power Analysis

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- a normal distribution with a mean equal to zero and a standard deviation that is equal to one. Used in hypothesis testing about means or proportion of samples drawn from a population whose population standard deviations are known

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Z distribution

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a continuous probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown… the distribution is symmetric and bell shaped with heavy tails

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t distribution

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a statistical test to determine whether a sample mean is different from a given µ... A z test is an statistical test for which the distribution of the test of the test statistic under the null hypothesis can be approximated by a normal distribution…must know sigma

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One-mean Z test

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a test to determine whether does the mean of a normally distributed population have a value specified in a null hypothesis. You can use when you have interval or ratio level data from a single sample of subjects. You want to determine whether the mean from the sample is significantly different from a specified population mean.

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One-mean t test

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the number of scores whose values are free to vary

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Degrees of Freedom

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can be used when the observations are independent. Observations are independent if the possibility of a specific score occurring under one condition is not influenced by scores occurring under another condition.

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Independent-means t test

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- a test of differences between 2 means… Use this when you want to compare average scores on a criterion variable under 2 conditions to determine if there is a significant difference between the two means. One set of scores is paired in a meaningful way with another set of scores.

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Correlated-means t test

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- (just because a difference is statistically significant doesn’t mean that it is big, important or helpful in decision making)… a measure of the strength of a relationship between 2 variables to determine whether the difference is meaningful

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Effect size

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of an Independent Means T Test
Homogeneity of Variance- is the variance the same or close enough?
Independence of Observations- determine from the study design
Normality of Population Distribution- check using plots, values of skewness and Kurtosis, Proc Univariate Normal, Shapiro-Wilk test or Kolmogorov-Smirnov Test.
Dependent Means only needs normality of population distibutions.

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Assumptions (VIN)—

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the extent to which the assumptions may be violated without distorting Type I error control… robustness properties normality, homogeneity of variance, independence. ???

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Robustness

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a test made to determine whether several populations are similar or equal or homogeneous in some characteristics… evaluates the equality of several populations of categorical data. The test asked whether 3 or more
populations are equal with respect to some characteristics.

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χ2 Test of homogeneity/independence

The only difference between the test for independence and the homogeneity test is the stating of the null hypothesis:

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This test is commonly used to test association of variables in two-way tables where the assumed model of independence is evaluated against the observed data. Describes how well a model fits a set of observations.

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χ2 Goodness of fit test

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any relationship between two measured quantities that renders them statistically dependent

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Measures of Association