Basic Statistics(12)-Hypotheses and significance tests

 
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Basic Statistics - (12) Hypotheses and Significance Tests

1. Hypotheses(由样本出现概率反推假设是否成立)

  • expectations about population

  • An hypothesis is formulated as a claim that a population parameter takes a particular value or falls within a specific range of values.

  • Null-hypothesis testing

    • H0: null hypothesis
      • the parameter you’re interested in takes a specific value.
      • will be rejected if the data in your sample suggest that it is a highly unlikely expectation.
    • Ha: alternative hypothesis
      • the parameter you’re interested in falls within an alternative range of values.
    • failing to reject your null hypothesis DOESN’T mean that the null hypothesis is true.

2. Proportion hypothesis（z-score）

  • Significance level(α):
    • one-tail:0.05
    • two-tail:0.025 (most)

3. Step by step plan

  • One warning:

4. Mean Significane test and Confidence interval

  • 2 methods of Inferential statistics

    • confidence intervals \(\\ \overline{x}\pm{t_{95\%}}(SE) \\where \ SE = \frac{S}{\sqrt{n}}\)

    • significance tests

      \(\\ t = \frac{\overline{x}-\mu_0}{SE} \\where \ SE = \frac{S}{\sqrt{n}}\)

  • the conclusion between CI and HT

5. Type I and Type II error

  • Type I

    • H0 为真，却被拒绝。
    • 原因
      • 选取样本太过极端会导致这种错误。（p<0.5）
    • 解决方案
      • Significance level(α)减小，作出错误判断的概率越小。

  • Tpye II

    • H0为假，却被接受。
    • 选取假设太过极端会导致这种错误。

  • The relation between Tpye I and Type II

  • Solution

    样本量的增加会降低标准误的大小（标准误=S/根号N，样本本量N越大，标准误越小，反映在图形中就是两个总体（假设总体和实际总体）变得更“细瘦”，所以重合的部分越少，由此代表犯错概率的图形的面积也会变小。

6. Example

  • Notes:

    1. always check the assumption
    2. always draw the sample distribution and check the critical value.

7. Limitations of ST

  • When a P-value is small, the significance test indicates that the hypothesized value is not plausible, but it tells us little about which potential parameter values are plausible.
  • “Do not reject H0” does not mean “Accept H0.”
  • Statistical significance does not mean pratical significane.
    • A small P-value does not tell us if the parameter value differs by much in pratical terms from the value in H0.
  • The P-value cannot be interpreted as the probability that H0 is true.
  • It is misleading to report results only if they are “statistically significant.”
  • Some tests may be statistically significant just by chance.
  • True effects may not be as large as initial estimates reported by the media.