A sample of 25 bottles is taken from the production line at a local bottling plant. Assume that the fill amounts follow a normal distribution. What is the probability that the sample standard deviation is more than 70% of the population standard deviation?

Question

BlackTom AI · Accepted Answer

For a normal population, (n − 1)S^2/σ^2 has a chi-square distribution with ν = n − 1 degrees of freedom. This key result lets us convert a probability about the sample standard deviation S into a probability about a chi-square random variable.

Step-by-step reasoning:
1. Identify parameters and target: n = 25, so ν = 24. We want P(S > 0.7σ). Equivalently P(S/σ > 0.7).
2. Square both sides to work with variances: P(S^2/σ^2 > 0.49).
3. Multiply by (n − 1) to use the chi-square relation: (n − 1)S^2/σ^2 = 24 · S^2/σ^2. So the event S^2/σ^2 > 0.49 is equivalent to (n − 1)S^2/σ^2 > 24 × 0.49.
4. Compute the threshold: 24 × 0.49 = 11.76. Thus P(S > 0.7σ) = P(χ^2_24 > 11.76).
5. Evaluate the chi-square tail probability: using a chi-square table or calculator, the cumulative probability F(11.76; ν = 24) ≈ 0.143, so the upper-tail probability is 1 − 0.143 = 0.857.

Intermediate calculations shown:
- ν = n − 1 = 24
- (S/σ)^2 threshold = 0.49
- χ^2 threshold = 24 × 0.49 = 11.76
- P = P(χ^2_24 > 11.76) = 1 − F(11.76;24) ≈ 1 − 0.143 = 0.857

Why this answer is correct:
The chi-square relation requires normality; with that assumption the conversion and threshold are exact. The computed tail probability 0.857 is the probability that the sample standard deviation exceeds 70% of the population standard deviation for n = 25.

Why the other options are incorrect:
- 0.157 is the complement (1 − 0.857); it would be P(S ≤ 0.7σ), not the probability asked.
- 0.542 and 0.982 do not match the chi-square tail probability for the computed threshold and so are incorrect.

Final confirmation: The correct probability is 0.857, matching the provided answer.

A sample of 25 bottles is taken from the production line at a local bottling plant. Assume that the fill amounts follow a normal distribution. What is the probability that the sample standard deviation is more than 70% of the population standard deviation?单项选择题

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