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Test 4 - Coggle Diagram

- - - - Change sample size N
      - Edit proportion
    - - If my sample size is not large, I want to make sure that the proportion is close to center
      - Large sample size is not necessary but helpful
    - - The sampling distribution for p̂ is approximately Normal if both np̂ and n(1-p̂) are at least 10
      - ==> "sufficiently large"
    - - For random samples with a sufficiently large sample size, the distribution of many common sample statistics can be approximated with the Normal distribution
      - = N (Null parameter, SE)
  - - - 1.96 = z critical value for 95%
      - z* = z-critical value
      - CI: 90% z*=1.645
      - CI: 99% z*= 2.575 or 2.576
  - - - 2) Calculate np̂ and n(1-p̂)
      - 3) Is each of them 10?
      - 4) Find z* for ...% confidence interval (1.645 for 90%; 1.96 for 95%; 2.576 for 99%)
      - 5) Calculate ME using z* x SE (give ME to one decimal place further than proportion p̂)
      - 6) Find the confidence interval using sample statistic p̂ +/- ME
      - 1) Identify p̂ and n
    - - Round SE to at least 4 decimal places
  - - - If historic data are applicable, use p̂ = p̃
      - If not, use the conservative estimate p̃=0.5
  - - - P0 (null-proportion) is the value that shows up in the Null-Hypothesis
      - Checking conditions
        
        We are checking if the distribution is Normal with np̂ and n(1-p̂) are at least 10
        
        All calculations in a hypothesis test are based on the assumption that H0 is true and therefore distribution we use for everything is centered at the null-proportion (p0) not at the sample proportion.
      - Example
        
        1) State Hypotheses H0 and Ha
        
        2) Calculate sample statistic
        
        3) Verify conditions for using a Normal distribution (np̂ and n(1-p̂) are at least 10)
        
        4) Calculate SE using p0
        
        5) Calculate z using the formula with p̂ and p0
        
        6) Show z on the Normal curve (how many standard deviations from 0 is it away)
      - In Statkey: Theoretical Distributions --> Normal
  - - - Calculate if np̃ and n(1-p̂) are at least 10
    - - With ...% confidence, between ... and ... (confidence interval) of all (population) (response variable)
    - - Bootstrap distribution for one proportion (percentiles method)
- - - - The sampling distribution gets closer to bell-shaped
      - The SE gets smaller
  - - - n-1 independent terms, therefore n-1 degrees of freedom
    - - If n is at least 30 we generally assume we can use a t-distribution for inference
      - If n is < 30 we want the sample to be fairly symmetric and unimodal (more so for smaller values of n)
      - If the values are clustered together when n<30 we should be able to use a t-distribution (it is hard to get a nice bell-shape with little values)
      - If n<30 (or 30) and we have an extreme outlier we will not use the t-distribution
  - - - s = Standard Deviation (read from Statkey)
  - - - OR Statkey: Theoretical Distributions --> t --> input degrees of freedom (n-1) --> select two-tail --> write 0.025 into top left or right numbers
    - - 1) ME = t* x SE
      - 2) x̅ +/- ME
- - - - 1) State Hypothesis (in both ways)
      - 2) Calculate the sample statistic
      - 3) Use the SE applet to calculate SE (use hypothesis test option)
      - 4) Calculate the z-statistic
      - 5) Find the p-value in Statkey (Theoretical Distributions --> Normal)
      - 6) State the conclusion using non-technical language (= complete the hypothesis test using a Normal distribution)
  - - - Each count needs to be at least 10
      - Or successes and failures of both variables need to be above 10
  - - - 1) Find sample statistic, SE and z* (look up SE in applet using CI)
      - 2) Calculate ME (z* x SE)
      - 3) Find CI by calculating p̂1-p̂2 +/- ME
  - - - Compare the proportion of members of party A who support lamps to the proportion of members of party B who support lamps
      - Compare the proportion of pet owners in ATX who buy food online to the proportion of pet owners in NYC who shop online
    - - Among all members of party A, how many support lamps and how many don't support lamps?
      - Compare the proportion of people in group A who buy fish to the proportion of people in the same group who buy meat
- - - - Best for Matched Pairs!
      - Ok for matched pairs or two independent samples
    - - We need sample size, mean and standard deviation
      - 1) calculate t-statistic
      - 2) Go to Statkey --> t-distribution
      - 3) put in n-1 degrees of freedom
      - 4) Choose correct tail
      - 5) Change t-statistic to what we've just calculated (number on x-axis)
      - 6) Read p-value
      - 7) Interpret the p-value
  - - - Preferred for matched pairs
      - Also correct for matched pairs and two samples
    - - 1) Find ME
      - 2) Write down CI (... , ...)
      - 3) Interpret the CI
- - - - We take the smaller n and subtract 1
    - - For using t-distribution both values of n must be > 30
      - When using the applet and my exact n isn't provided, just use the closest one
      - If the p-value from the t-distribution is very far from the p-value gotten from the randomization distribution, the latter is correct.
      - If n>30 with skew we can use t distribution. If we have an extreme outlier, we cannot