---
title: "Confidence level; bulb superiority claim"
subject: "Probability and Statistics"
chapter: 6
key: "semester_5/probability_and_statistics:confidence-level-bulbs-superior-claim"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/confidence-level-bulbs-superior-claim
site: BCT Engineering
---

# Confidence level; bulb superiority claim

Probability and Statistics · Chapter 6: Inference concerning mean

- Asked in: 2071 Chaitra, 5 marks
- Video: [Confidence Interval vs Confidence Level || The difference and how to use them](https://www.youtube.com/watch?v=oaXX_Sn8FOo) — Science Grad School Coach
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/confidence-level-bulbs-superior-claim
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6

## Question

Define confidence level and significance level. A company claims that its light bulbs are superior to those of its main competitor. If a study showed that a sample of 40 of its bulbs has mean lifetime of 647 hours of continuous use with standard deviation of 27 hours, while a sample of 40 bulbs made by its main competitor had mean lifetime of 638 hours of continuous use with standard deviation of 31 hours. Does this substantiate the claim at 1% level of significance?

## Short answer

Confidence level (1−α) is the probability an interval contains the parameter; significance level α is Type I error probability. Right-tailed z-test: SE=√(27²/40+31²/40)=6.5, z=1.385 < critical 2.326; fail to reject H0, the claim is not substantiated at 1%.

## Answer

#### Confidence level and significance level

- **Confidence level (1 − α)** is the probability (usually 90%, 95% or 99%) that a confidence interval, constructed by the given method, contains the true population parameter. A 95% level means that 95 out of 100 such intervals would capture the parameter.
- **Significance level (α)** is the maximum probability of rejecting a true null hypothesis (Type I error), fixed before the test, commonly 0.05 or 0.01. It is the size of the rejection region. Confidence level = 1 − α.

#### Test of the company's claim

The company (sample 1) claims superiority, so a right-tailed z-test is used (both samples large).

1. H0: μ₁ = μ₂ (the company's bulbs are not better)
   H1: μ₁ > μ₂ (the company's bulbs last longer) (right-tailed)
2. Level of significance: α = 0.01
3. Test statistic (large samples, z-test):

```text
z = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)
SE = √(27²/40 + 31²/40) = √(18.2250 + 24.0250) = 6.5000
z  = (647 − 638) / 6.5000 = 1.385
```

4. Critical value: z at α = 0.01 (right-tail) = 2.326. Reject H0 if z > 2.326.
5. Decision: the calculated value falls in the acceptance region (p-value = 0.0831), so **fail to reject H0**.

**Conclusion:** the sample does not substantiate the claim at the 1% level; the observed 9-hour difference may be due to sampling variation.

## Check yourself

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

1. What is a 95% confidence level?
   A. The probability H0 is true
   B. A method for which 95 of 100 such intervals capture the true parameter
   C. 95% of the sample lies in the interval
   D. The probability of a Type I error
   Answer: B. Confidence level (1 - alpha) refers to how often intervals built this way contain the parameter. (Key point: Confidence level)

2. What is the significance level alpha?
   A. The size of the sample
   B. The maximum probability of rejecting a true H0 (Type I error)
   C. The probability of accepting a false H0
   D. 1 minus the p-value
   Answer: B. Alpha is the maximum Type I error probability and the size of the rejection region. (Key point: Significance level)

3. Why is the test right-tailed?
   A. H1: mu1 > mu2 because the company claims its bulbs are superior
   B. H1: mu1 < mu2
   C. Because n is large
   D. H1: mu1 != mu2
   Answer: A. The claim of superiority is directional, giving a one-tailed test. (Key point: Alternative hypothesis)

4. What is the decision at 1%?
   A. z = 1.385 > 2.326, reject
   B. z = 6.5, accept the claim
   C. z = 1.385 < 2.326, fail to reject H0; claim not substantiated
   D. z = 9 so reject
   Answer: C. The critical value is 2.326 and 1.385 is below it. (Key point: Decision)

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