---
title: "CI: mean pH in rainfalls"
subject: "Probability and Statistics"
chapter: 6
key: "semester_5/probability_and_statistics:ph-rainfalls-ci-99"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/ph-rainfalls-ci-99
site: BCT Engineering
---

# CI: mean pH in rainfalls

Probability and Statistics · Chapter 6: Inference concerning mean

- Asked in: 2070 Chaitra, 4 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/ph-rainfalls-ci-99
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6

## Question

An analysis for pH (acidity) in a random sample of water from 40 rainfalls showed that mean is 6.7 and s.d. is 0.5. Find a 99% confidence interval for the mean pH in rainfalls.

## Short answer

With n=40, use normal z-interval: x̄ ± 2.576·s/√n. SE=0.0791, margin=0.2036. 99% confidence interval for mean pH is (6.50, 6.90).

## Answer

The sample is large (n = 40), so the normal interval x̄ ± z·s/√n is used, with s as the estimate of σ.

```text
n = 40, x̄ = 6.7, s = 0.5
SE = s/√n = 0.5/√40 = 0.0791

99% CI: z = 2.576
  margin = 2.576 × 0.0791 = 0.2036
  limits = 6.7 ± 0.2036 = (6.496, 6.904)
```

**Answer: 99% confidence interval for mean pH = (6.50, 6.90).**

## Check yourself

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

1. Why use z?
   A. n = 40 is large; s estimates sigma
   B. sigma is unknown and n tiny
   C. n is small
   D. Data are paired
   Answer: A. Large sample so normal critical values are used. (Key point: Distribution)

2. What is SE?
   A. 0.5/sqrt(40) = 0.0791
   B. 0.5^2/40
   C. 0.5 x sqrt(40)
   D. 0.5/40
   Answer: A. SE = 0.5/6.325 = 0.0791. (Key point: Standard error)

3. Which z is used for 99%?
   A. 2.326
   B. 1.645
   C. 1.96
   D. 2.576
   Answer: D. 99% two-sided uses 2.576. (Key point: Critical value)

4. What is the interval?
   A. (6.50, 6.90)
   B. (6.62, 6.78)
   C. (6.2, 7.2)
   D. (6.7, 6.9)
   Answer: A. 6.7 +- 2.576 x 0.0791 = 6.7 +- 0.204. (Key point: Result)

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