Statistics and Probability

• Hypothesis testing - parameter is tested using sampling data
• Hypothesis - assumption for the sake of argument; educated guess
• Null Hypothesis - no difference
• Alternative Hypothesis: statement of difference & contains equality
• Null: =, ≥, ≤

  Alternative: ≠, <, >
• Types of AH:

  1. Directional AH: asserts one measure is >, <
  2. Non-Directional AH: unequal; ≠
• Hypothesis: direction to the researcher's thinking about the problem, and therefore, facilitates a solution.
• Forms of Statistical Tests:

  1. One-Tailed test: null against directional
  2. Two-tailed test: null against non-directional; one measure is different from another
• Two types of One-tailed test:

  • Left-tailed: one measure is <
  • Right-tailed: one measure is >
• Central Limit theorem - identifies appropriate statistical test
• Z-test: known psd, large sample s (n≥30)

  T-test: unknown psd, small sample s
• Steps in Hypothesis testing:

  1. Formulate the hypotheses
  2. Collect data and describe
  3. Specify the level of significance and compute test-stats
  4. Determine the critical region
  5. Decide and Conclude
• Errors:

  Type I: rejecting null H when it is true

  Type II: favoring the null H when it is false
• Probabilities:

  • P(Type I) = alpha
  • P(Type II) = beta
• Level of significicance:

  • education = 0.05
  • medicine = 0.01
  • other uses = 0.10
• Computed Measures:

  1. Test statistic: computed from sample data; concerned with values (z, t, etc.)
  2. Significance Probability: the probability using the test statistic; probability
• Test Value = observed value - expected value / standard error
• Regions of Distribution:

  1. Rejection region: supports AH tat rejects the null
  2. Non-rejection region: values that support the null
• Two-tailed test: A= a/2; A= 1-a

  One-tailed test: A=a; A= 1-a
• If pasok, reject

  If not pasok, do not reject
• Ho: Reject, There is enough evidence to reject the claim

  Do not reject, There is no enough evidence to "

  H1: Reject: There is enough evidence to support the claim

  Do not reject, There is no enough evidence to "
• One Population test - conducted on one sample from population with a mean; a.k.a. Significance Test for a single mean. Tests that the unknown population is equal to the hypothesized population
• Large sample test: using a large sample

  Small sample test: using a small sample
• Hypothesis testing two independent population - using two sample from population to compare to unknown population
• Independent population - no relations

  Dependent population - has relations; a.k.a. paired/related; if matched/paired
• Two Independent sample test: when two indp sample are drawn from normal population with known psd; for z-test
• Homogenety of Variance (T-test):

  • Pooled variance: equal
  • Separate Variance: unequal
• Independents samples:

  • Z-test for two indp sample means
  • Pooled variance t-test (equal)
  • Separate variance t-test (unequal)
• Dependent Samples:

  • Paired samples
  • Matched samples
  • Related samples
• Paired samples: data value collected has a corresponding data value from second sample; both are collected/related/matched from same source
• Paired Sample test: t -test