Hypothesis testing and contextual conclusions
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| significance level/sɪɡˈnɪfɪkəns ˈlevl/ | mức ý nghĩa |
Could chance explain the result?
- A factory claims that only 10% of items are defective. A sample contains more defects, but chance alone may explain some difference.
- This lesson studies significance level 显著性水平: The chosen probability threshold for rejecting a null hypothesis.
The method
- State H₀ and H₁ in population parameters before inspecting the outcome.
- Calculate the appropriate tail probability under H₀.
- Reject H₀ when the evidence meets the specified significance rule; otherwise say there is insufficient evidence.
At the 5% level the p-value is 0.07 for H₀:p=0.1. What is the correct conclusion?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
0.07 exceeds the 0.05 threshold, so the data do not meet the significance rule. That is not proof H₀ is true - the evidence is simply too weak against it.
A worked case
For H₀:p=0.1 and H₁:p>0.1 with n=10, observing at least 3 defects has probability 1-(0.9^10+10×0.1×0.9^9+45×0.1²×0.9^8)≈0.070191. At 5%, this is insufficient evidence that the defect rate exceeds 10%.
Hypothesis testing and contextual conclusions
State H₀ and H₁ in population parameters before inspecting the outcome
Compare the model with the worked case and explain one change.
Find the significance threshold written as a decimal for 5%.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
5% means 5/100=0.05.
Where it goes wrong
- Failing to reject H₀ is not proof that H₀ is true.
- Choose the tail from H₁, not from whichever tail gives a small result.
- Statistical significance does not measure the practical size of an effect.
Not true: Failing to reject a null hypothesis proves it is true. Not rejecting the null hypothesis only means there is not enough evidence against it.
A correct one-tailed p-value is 0.03 at a 5% threshold. Enter 1 for reject or 0 for do not reject.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The correct tail probability 0.03 is below 0.05, so reject H₀. This is evidence, not proof.
Not rejecting the null hypothesis only means there is not enough evidence against it.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Not rejecting the null hypothesis only means there is not enough evidence against it.
Applying it
- Finish with a sentence about the population and the original claim.
- State the model's assumptions and consider whether the sampling procedure supports them.
Find P(X≥3) for X binomial(10,0.1). Give it to 3 significant figures.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Sum the binomial probabilities for k=3 through 10; equivalently use 1-P(X≤2)=0.0701908264.
Match each testing term to its role.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Hypotheses are stated before inspecting data, and the tail follows H₁; the p-value is compared with the pre-set significance level.
In your course
- Pearson Edexcel International A Level Mathematics, unit S2.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Significance level: The chosen probability threshold for rejecting a null hypothesis.
- State H₀ and H₁ in population parameters before inspecting the outcome.
- Watch for: failing to reject H₀ is not proof that H₀ is true.
Write from a p-value of 0.08
- The test is at the 5 percent level.
- The p-value is 0.08.
- Say whether to reject.
A one-tailed test uses a 5 percent significance level and the p-value is 0.08. Should the null hypothesis be rejected?
0.08 is above the 5 percent threshold.
What is the decision?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A larger p-value than the threshold does not reject.