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Statistics (Binomial Probability Distribution (Rules (1) must have a fixed…

- - - - n!
        
        the number of ways to arrange 'n' number of unique items
      - nPr = n! / (n-r)!
        
        to only permute a select few of the total 'n' items. you don't need them all
    - - 1) the items have to be unique
      - 2) r is the number you are selecting to arrange out of 'n' unique items.
      - 3) order or arrangement MATTERS
- - - - this mean is also known as the "Expected Value" because the mean of the probability distribution tells you what x values are most probable in the distribution
    - - sigma^2 = sum[ x^2 * P(x)] - mu^2
    - - It tells you the average distance from the mean; it is expressing on average how far all the data points are from the mean of the those data points
    - - Values that are within 2 standard deviations of the mean
        
        mu + 2*sigma
        
        mu - 2* sigma
        
        Based on the empirical rule, values that are greater than 2 standard deviations from the mean will have 5% or less probability of occurring
- - - - mu = n * p
- - - - Special note to self: the reason for the cumulative distribution table, is that someone has already calculated the area under the curve for the standard normal distribution. If you have the table all you have to do is use the table to look up my Z score and then find the area I want. or if I have an area then I can find the Z score associated with it.
      - There are many different normal distributions but by mapping any normal distribution to a standard normal then you can use the table to look up areas and find the probabilities of certain outcomes occurring.
- - - - How about finding the "center" of the data, the middle or most common element in my data set. What is occurring most often?
        
        This is Central Tendency
  - - - Describing variability
        
        Range
        
        The difference between the minimum and maximum value of the dataset
        
        Variance
        
        its the average of the squared differences FROM the mean . it represents the total distance of the data from the mean.
        
        Standard Deviation
        
        It's the square root of the Variance. A small standard deviation indicate that most of the data points are near the mean, but a large standard deviation indicate that the data is scattered far from the mean
- - - - example: if you take a sample size of greater than 30, calculate the mean for each sample, put that collection of sample means into a table and the distribution of those means will be normal
      - When taking the average of the sample means taken from all the possible samples in a population of size N, it equals the population mean. mean of sample means = mu
      - taking the standard deviation of all possible sample standard deviations does NOT equal the population sigma, though! it equals this: st.dev of sample st. dev = sigma/sqr(n) It will be and underestimate because as the sample size of n gets bigger the st. dev of the sample st. dev will get smaller. Some of the data gets lost. prof leonard explans this in this section. this formula is also called the Standard Error
      - note if the sample size is less than 30, the formulas written here are always true and correct, however the assumption that the sampling distribution is normally distributed is predicated on n being greater than 30!!! If the sample size is less than 30 and you don't know what the underlying population distribution looks like then you can't assume that the sampling distribution is normal
  - - - note that assumptions have to be made about mu and sigma in order to apply this formula
- - - - 1) must have a random sample
      - 2) conditions for a binomial = having a fixed number of trials, trials are independent, 2 outcomes of success or failure, n*p >= 5, nq >= 5
    - - tells you how sure you are that the actual population parameter is in the range given
        
        Confidence level = 1 - alpha
        
        Most common confidence Levels: 90%, 95%, 99%
      - alpha: is the complement of a Confidence Level
    - - What is a critical value? it is a Z score that separates the likely region from the unlikely region . it is the z score that corresponds to the alpha/2 areas under the normal distribution curve
    - - qbar = the sample proportion of failure
        
        qbar = 1 - pbar
      - P = population proportion of sucess
      - point estimate = one value that you used to represent a population
      - pbar = sample proportion of sucess
        
        pbar = x/n
        
        where x is the number of successes out of 'n' trials
- - - - 1) is the problem talking about a proportion or a mean ?
      - 2) is the problem need a z score or a t-score? this is determined by knowing of the population st. dev is know or not. If population st. dev is known, then use z score, but if sigma is unknown use t score/ t -test