We study the idea of statistical inference: “Set a null hypothesis (H₀) and make a judgment based on the data.” (In Mathematics I, the focus is mainly on intuitive understanding.)
Process of Hypothesis Testing
① Set the null hypothesis H₀ (“no effect,” “no difference,” etc.)
② Evaluate how likely the observed data are
③ If the result is unlikely (less than 5%, etc.), reject H₀
Example ①
A coin was tossed 10 times, and heads appeared 9 times. Is this coin fair?
Answer: H₀: The coin is fair (heads and tails each have probability 1/2). The probability of getting heads 9 or more times in 10 tosses is approximately 1.1% (small). Reject H₀ → It is highly likely that the coin is not fair.
A coin was tossed 10 times, and heads appeared 9 times. Is this coin fair?
Answer: H₀: The coin is fair (heads and tails each have probability 1/2). The probability of getting heads 9 or more times in 10 tosses is approximately 1.1% (small). Reject H₀ → It is highly likely that the coin is not fair.
Key Point
- Judgments are often made using a significance level of 5% (0.05) as the standard
- Rejecting ≠ “definitely different” (it is a probabilistic judgment)
Practice Problems
- A die was rolled 12 times, and the face 1 appeared 5 times. Is this die fair? (Describe the reasoning for making a judgment at a significance level of 5%.)
Answers and Explanations
- Answer: If the die is fair, the expected number of 1s is 2. Five is more than expected, but calculate the probability of getting 5 or more 1s in 12 rolls and compare it with 5%. (The calculation is advanced.)