---
title: "Confidence level; cables conforming claim"
subject: "Probability and Statistics"
chapter: 7
key: "semester_5/probability_and_statistics:confidence-level-cables-95-claim"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/confidence-level-cables-95-claim
site: BCT Engineering
---

# Confidence level; cables conforming claim

Probability and Statistics · Chapter 7: Inference concerning proportion

- Asked in: 2071 Shrawan, 5 marks
- Video: [Hypothesis Testing - Solving Problems With Proportions](https://www.youtube.com/watch?v=76VruarGn2Q) — The Organic Chemistry Tutor
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/confidence-level-cables-95-claim
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-7

## Question

Define confidence level and significance level. A manufacturer claimed that at least 95% of the cables supplied to the ABC Company confirmed to specifications. However, the production manager at ABC Company wasn't satisfied with the claim of the manufacturer. Hence, to test the claim, the manager examined a sample of 250 cables supplied last month and found that 228 cables as per the specifications. Can you conclude that the production manager is right to doubt on the claim of the manufacturer? (α = 0.01)

## Short answer

Confidence level is 1-alpha, the probability an interval contains the parameter; significance level alpha is P(Type I error). For H0:p=0.95 versus H1:p<0.95, z=-2.757; critical value at 0.01 is -2.326, so reject H0. The manager is right to doubt the claim.

## Answer

**Confidence level:** the probability (1 − α), such as 95% or 99%, that an interval estimate will contain the true population parameter if sampling is repeated many times.
**Significance level (α):** the probability of rejecting a true null hypothesis (Type I error), such as 5% or 1%. The confidence level equals 1 − α.

#### Test of the claim

The claim is p ≥ 0.95, so the doubt is that p < 0.95 (left-tailed test, large sample).

```text
n = 250, x = 228, p̂ = 228/250 = 0.912,  p₀ = 0.95, q₀ = 0.05
```

1. H0: p = 0.95 (the claim holds, at least 95% conform)
   H1: p < 0.95 (fewer than 95% conform; the manager's doubt)
2. Level of significance: α = 0.01
3. Test statistic:

```text
SE = √(p₀q₀/n) = √(0.95×0.05/250) = √0.00019 = 0.013784
z  = (p̂ − p₀)/SE = (0.912 − 0.95)/0.013784 = -2.757
```

4. Critical value: z(0.01, left-tail) = −2.326. Reject H0 if z < −2.326.
5. Decision: z = -2.757 < −2.326 (p-value = 0.0029), so **reject H0**.

**Conclusion:** at the 1% level the sample does not support the manufacturer's claim. The production manager is right to doubt that at least 95% of the cables conform to the specification.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the confidence level in terms of alpha?
   A. alpha
   B. alpha/2
   C. 1 − alpha
   D. 2alpha
   Answer: C. Confidence level = 1 − alpha, e.g. 99% for alpha = 0.01. (Key point: Confidence level)

2. What is the significance level?
   A. Power
   B. Probability of accepting a true H0
   C. Probability H0 is true
   D. Probability of rejecting a true H0
   Answer: D. alpha is the probability of a Type I error. (Key point: Significance level)

3. What is H1 for the manager's doubt?
   A. p < 0.95
   B. p > 0.95
   C. p ≠ 0.95
   D. p = 0.95
   Answer: A. Doubting an 'at least 95%' claim: H1: p < 0.95. (Key point: Direction)

4. Critical value is −2.326. For the answer's z, what is the decision?
   A. Fail to reject H0
   B. Reject H0; the manager is right to doubt
   C. Accept the claim
   D. Two-tailed needed
   Answer: B. z is below −2.326, so H0 is rejected. (Key point: Decision)

## Related in Probability and Statistics

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- [Test of proportion: 95% equipment claim](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aequipment-95-percent-conform-test) — Probability and Statistics, chapter 7
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