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Statistics revision (ANOVA (check table saved in pictures) (Sum of squares…

- - - - F-test
        
        If there's no treatment effect, then:
        
        H0: umsg = umse
        
        If H0 holds true, we expect the F-ratio/ F-distribution to take a value of almost 1
        
        Ha: umsg > umse
        
        If Ha holds true, we expect the F-ration/ F-distribution to be smaller than 1
        
        To know whether F is large enough to conclude that Ha is true, we look at two parameters: dfg (df of MSG) and dfe (df of MSE) and use table E
        
        In Table E, we use the rows to search dfe and in the columns for dfg, from which we will determine the critical F-value and can reject H0 or not. :red_flag:
        
        Multiple Comparisons
        
        When we find out a significant effect, we have to find out which means exactly aren't equal
        
        1 more item...
        
        We do this through a pairwise comparison --> ie. 3 conditions, do 3 two-sample t-tests etc.
- - - - T-tests are quite robust against non-normality of the population (except in outlier cases or string skewedness)
        
        Larger samples improve the accuracy of p-values and critical values from the t distribution when the population is normal
  - - - Having said this, as n increases s , s becomes closer to o and thus the probability distributions look more alike as there is less uncertainty
    - - s= square root of the total sum of (x - x bar) 2/n-1
  - - - Needed to study the data of a single group.
    - - The goal is to compare the means of the response variable in two groups (using inferential statistics)
        
        The two groups are a sample from a distinct population and are independent to each other
      - Hypotheses
        
        H0: u1-u2=0
        
        Ha: u1-u2 > 0
      - Standard error: x bar 1 - x bar 2
      - In the case when the standard deviation is the same in two populations o1=o2 we use Sp, the pooled sample standard deviation --> in SPSS assumes equal variances
        
        If o1=o2= o then we use the test-statistic: T=(x bar 1 - x bar 2) - (u1 - u2) / square root of MSE and square root of 1/n1 + 1/n2
        
        In order to calculate Sp: (square root of ) s2p= (n1-1) s21 + (n2 -1) s22/(n1-1) + (n2-1)
        
        To know whether o1=o2 there's a rule of thumb
        
        If largest s > 2* smallest we don't assume it : :!?:
        
        When both n are almost equal
      - df= (n1-1) + (n2-1) = n1 + n2 -2
    - - Similar to one-sample t-tests but we compute the difference between two groups instead
- - - - Sample proportion of success is used to estimate the unknown population proportion (p)
  - - - H0: our observed proportions are the same as the hypothesised proportions
        
        We have to find out what the expected count is in order to find out whether our null hypothesis is true or not.
        
        Expected count (EC): n* pi (expected proportion for one cell)
        
        Chi-square statistic test compares the observed counts with the expected counts:
        
        Under H0, X has around the distribution X with df (number of categories take away 1)
        
        Then we can determine using the critical X^2 value and the X^2 distribution table (Table F) how significant our result is
      - Ha: our observed proportions differ from the hypothesised proportions
  - - - Again we have to find out whether the null hypothesis is true through the expected cell counts, this time it has a different formula: EC= total row x total column/ n
        
        All expected counts should be 5 or greater
      - Chi-square statistic test compares the observed counts with the expected counts and has the same equation as shown below
      - Under H0, X^2 has approximately the X^2 distribution with (row-1) (column-1) df.
      - The p-value for the test is P (X^2 distribution table > or equal to X^2 observed value).
        
        To obtain the p-value, we look at the X^2 distribution table (Table F)