乜野係......hypothesis testing

Hypothesis testing is the use of statistics to determine the probability that a given hypothesis is true. (For example, in the exam, to test whether the mean score of boys and girls are equal or not) The usual process of hypothesis testing consists of four steps.
1. Formulate the null hypothesis
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(commonly, that the observations show a real effect combined with a component of chance variation).
2. Identify a test statistic that can be used to assess the truth of the null hypothesis.
3. Compute the P-value, which is the probability that a test statistic at least as significant as the one observed would be obtained assuming that the null hypothesis were true. The smaller the
圖片參考：http://mathworld.wolfram.com/images/equations/HypothesisTesting/inline3.gif
-value, the stronger the evidence against the null hypothesis.
4. Compare the
圖片參考：http://mathworld.wolfram.com/images/equations/HypothesisTesting/inline6.gif
, that the observed effect is statistically significant, the null hypothesis is ruled out, and the alternative hypothesis is valid.
Common test statistics



Name
Formula
Assumptions

One-sample z-test

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(Normal distribution or n ≥ 30) and σ known

Two-sample z-test

圖片參考：http://upload.wikimedia.org/math/5/3/d/53d9cfd60853324980381ac8dbd09f38.png

Normal distribution and independent observations and (σ₁ AND σ₂ known)

One-sample t-test

圖片參考：http://upload.wikimedia.org/math/c/6/5/c659df151a26b32aa4e2be12f753b774.png

df = n − 1

(Normal population or n ≥ 30) and σ unknown

Two-sample pooled t-test

圖片參考：http://upload.wikimedia.org/math/e/8/f/e8f7dba8e6f56fb75d57e8952cbc7a4a.png


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df = n1 + n2 − 2

(Normal populations or n₁ + n₂ > 40) and independent observations and σ₁ = σ₂ and (σ₁ and σ₂ unknown)

Two-sample unpooled t-test

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or df = min{n1,n2}

(Normal populations or n₁ + n₂ > 40) and independent observations and σ₁ ≠ σ₂ and (σ₁ and σ₂ unknown)

Paired t-test

圖片參考：http://upload.wikimedia.org/math/c/d/4/cd4d7afafc0c609749f0a2e0f065d559.png

df = n − 1

(Normal population of differences or n > 30) and σ unknown

One-proportion z-test

圖片參考：http://upload.wikimedia.org/math/2/0/b/20bd40213fb213e51cc8e15633cf7e2e.png

np > 10 and n(1 − p) > 10

Two-proportion z-test, equal variances

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n₁p₁ > 5 AND n₁(1 − p₁) > 5 and n₂p₂ > 5 and n₂(1 − p₂) > 5 and independent observations

Two-proportion z-test, unequal variances

圖片參考：http://upload.wikimedia.org/math/5/b/a/5ba0665f67034488b35cda1127a9e23a.png

n₁p₁ > 5 and n₁(1 − p₁) > 5 and n₂p₂ > 5 and n₂(1 − p₂) > 5 and independent observations

Information:
a. population variance (sigma^2) = 0.25
b. sample variance (s^2) = 0.27
c. n = 41
d. degree of freedom (df) = 41 - 1 = 40
e. alpha = 0.05

Answer:
1. Hypothesis
Ho: sigma^2 is less than or equal to 0.25
Ha: sigma^2 is greater than 0.25

2. Test Statistics (X^2)
X^2
= (n - 1) * s^2 / sigma^2
= (41 - 1) * 0.27 / 0.25
= 43.2

3. Critical Value based on 5% significance level
Chi-square critical value (df = 40, alpha = 0.05) = 26.51

4. Decision
Since X^2 = 43.2 is greater than 26.51 = Chi-square critical vaule, there is enough evidence to reject the company claim.

5. Conclusion
We can reject the company claim (that the variance of the amount of fat in the whole milk processed by the company is no more than 0.25) at 5% significance level.