Stats 8 - Infrence for Categorical Data

Unbiased Estimate

• Consider a categorial variable with proportions p of some outcome we label “success” in the population

  • sampling distribution of phat is roughly Normal if n is large enough

Confidence Interval for P

• Take a random sampel of n observation form a population with unknown proportions “p” of “successes”

  • each observation is a success or not

  • Find the proportion phat of successes in the sample

• An approximated level C Confidence Interval for p has teh form

  • unbiased estimate (phat) +- margin of error (m)

Conditions for Interference - Assumption of z confidence interval for a proportion

1. Categorical data are random samples

   1. or unbiased sample in a randomized experiment form a larger population

2. The counts of success/failures in teh sample are ideally at least 15 for a Normally sampling distribution of phat

X² test for Two-Way tables [X² = test statistic]

• Two way tables with r rows and x columns summarizes 2 categorical variables in random data

  • H0: There is no asssociation between row & column variables (variables are independent)

  • Ha: H0 is not true

• When H0 is true, an expected count is computed for each cell in the two-way table of counts, based on marginal distributions.

Conditions for Inference: Chi Test for Two Way Tables

1. Data are in one or more random samples from larger populationsl; each individual falls into one cell

2. Expected counts are large enough

   1. All expected counts have numerical values >_ 1.0

   2. most expected counts have numerical values >_ 5.0

   3. No more than 1 smallish value for every 5 values

X² test for goodness of fit

• Consider a categorical variable with proportions p1,p2,…pk in the population for the k outcomes making up the variable

  • p1 + p2 +… pk = 1

  • H0: p1= p1H0, p2 = p2H0, pk = pKH0