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Ch.6 Probability/Proportions, Ch.7 Distribution of Sampling Means, Ch.8…

- - - - 34.13% fall between mean and z = 1; 13.59% fall between z =1 and z = 2; and 2.28% fall between z = 2 and z = 3
  - - - The same table (in Appendix B) is used for all normal distributions
        
        To find proportions using X, first transform to z score
        
        Used in inferential statistics by observing if there is an extreme value that has a low probability of occuring
- - - - Example
        
        what is the probability a sample of n = 16 have a score higher than 525 when µ = 500 and ϭ = 100?
        
        1st- distribution is normal bc population is normal
        
        2nd- distribution has M = 500 bc µ = 500
        
        3rd-
        for n = 16, standard error ϭm = 25 bc ϭm = ϭ / √n
        
        4th-
        locate position with z score; in this case with ϭm = 25 (used as regular ϭ), z score = +1
        
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  - - - The expected value of M = the mean of the sample means and is near or matching the population mean
      - Sampling Error
        
        Difference between statistic and parameter. Because the sample is not a perfect representation of population
        
        (not to be confused with Standard Error which describes the average distance from the sample mean to the population mean)
        
        natural discrepancies like this are present in all inferential statistics and are necessary to account for
        
        If the change occurring in research is within the sampling error, it is not evidence of an effective intervention
- - - - For variance (ϭ^2) = ϭ = √ϭ^2
        For standard error = ϭm = √ϭ^2 / n
- - - - Step 3-
        Collect data and compute statistics
        (and obtain z score)
        
        Step 4-
        Make a decision
        (reject or fail to reject H0)
        
        Determine is sample mean is in the critical region
        
        if so, H0 is false (rejected) and change has occurred. There is good evidence to support H1
        
        If sample mean in not in critical region (close to original mean), then "Fail to reject H0" which means treatment does not have an effect
        
        Errors
        
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        Distance between sample mean and population mean is most influential factor
        
        Also variability and number in sample
        
        High variability/standard deviation lowers chances of seeing patterns or finding treatment effect
        
        small sample size decreases z score
        
        Raw data is used to create the sample mean that will be compared to the original mean
        
        comparison uses z score
        (M - µ) / ϭM
        (µ is from H0)
        (ϭM = ϭ / √n)
        
        Z score = Test Statistic = single specific statistic used to test hypothesis
        
        large value of test statistic = large discrepancy in data (likely in critical region)
      - consider which sample mean is consistent with H0
        
        divide the distribution of sample means into two sections: likely for H0 and unlikely for H0
        
        exact values for high and low probability
        
        Alpha level = low probability (.001, .01, and .05 are common)
        
        extremely unlikely values in the tails of the distribution (Alpha) = critical region
        
        If research produces a sample mean in the critical region, H0 is rejected and H1 is accepted
        
        Alpha level is also the probability that a Type l error will occur
        
        How to select Alpha level
        
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        under control of researcher
        
        use Alpha level probability and unit normal table (ex. a = .05, unit normal table for .025 on each side)
    - - The alternative hypothesis (H1) states that there is a change, difference, or relationship from the treatment. The original mean changes
- - - - Critical region is entirely on one side instead of split
        
        This means we can reject H0 when differences between means are smaller than with two-tailed
- - - - sample size is not considered in Cohen's d
- - - - Probability of Type ll error is still p = β; so probability of the power (outcome 2) is p = 1 - β
        
        The same size (but in percentage) of the critical region if the test is a success