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Probability:
For a situation in which several different outcomes are…
Probability:
For a situation in which several different outcomes are possible, the probability for any specific outcome is defined as a fraction or a proportion of all the possible outcomes. If the possible outcomes are identified as A, B, C, D, and so on...
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Probability of A = number of outcomes classified as A / total number of possible outcomes (only accurate for random samples)
Distribution of sample means is defined as the set of Ms for all the possible random samples for a specific sample size (n) that can be obtained from a given population. According to the central limit theorem, the parameters of the distribution of sample means are as follows:
a.Shape: The distribution of sample means is normal if either one of the following two conditions is satisfied:
(1)The population from which the samples are selected is normal.
(2)The size of the samples is relatively large (around or more).
b. Central Tendency: The mean of the distribution of sample means is identical to the mean of the population from which the samples are selected. The mean of the distribution of sample means is called the expected value of M.
c. Variability: The standard deviation of the distribution of sample means is called the standard error of M
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Frequency Distributions
probability questions are answered by determining proportions of area
Ex. The “probability of selecting an individual with an IQ greater than 108” is equivalent to the “proportion of the whole population that consists of IQs greater than 108.”
Normal Distributions
probabilities (proportions) can be found in the unit normal table.
The table provides a listing of the proportions of a normal distribution that correspond to each z-score value
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Binomial Distribution
used whenever the measurement procedure simply classifies individuals into exactly two categories. The two categories are identified as A and B, with probabilities of
p(A) = p
and
p(B) = q
gives the probability for each value of X, where X equals the number of occurrences of category A in a series of n events.
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Statistical inference involves using sample statistics to make a general conclusion about a population parameter. This is why standard error plays a crucial role in inferential statistics
Hypothesis testing is an inferential procedure that uses the data from a sample to draw a general conclusion about a population. The procedure begins with a hypothesis about an unknown population. Then a sample is selected, and the sample data provide evidence that either supports or refutes the hypothesis.
Four-step process
State the null hypothesis(Ho) and select an alpha level. The null hypothesis states that there is no effect or no change. In this case, Ho states that the mean for the treated population is the same as the mean before treatment. The alpha level provides a definition of the term very unlikely and determines the risk of a Type I error. Also state an alternative hypothesis H1, which is the exact opposite of the null hypothesis.
Locate the critical region. The critical region is defined as sample outcomes that would be very unlikely to occur if the null hypothesis is true
Collect the data, and compute the test statistic. The sample mean is transformed into a z-score
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Make a decision.
If the z-score is not in the critical region, fail to reject Ho because the data are not significantly different from the null hypothesis. In this case, the data do not provide sufficient evidence to indicate that the treatment has had an effect.
If the obtained z-score is in the critical region, reject Ho because it is very unlikely that these data would be obtained if Ho were true. In this case, conclude that the treatment has changed the population mean
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Power of a hypothesis test is the probability that the test will correctly reject the null hypothesis
First identify the treatment and null distributions. Also, you must specify the magnitude of the treatment effect.
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