---
title: "Test: soft drink filling machines"
subject: "Probability and Statistics"
chapter: 6
key: "semester_5/probability_and_statistics:soft-drink-machines-same-volume"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/soft-drink-machines-same-volume
site: BCT Engineering
---

# Test: soft drink filling machines

Probability and Statistics · Chapter 6: Inference concerning mean

- Asked in: 2068 Chaitra, 6 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/soft-drink-machines-same-volume
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6

## Question

A soft drink is being bottled using two different filling machines. The standard deviation of the process for machine A and B was 0.010 and 0.015 L respectively. 30 bottles were randomly sampled from each machine and the means were 2.04 and 2.07 L respectively. Can one conclude that both machines are filling the same volume of soft-drink? Test the hypothesis at α = 0.01 level of significance.

## Short answer

Two-sample z-test gives z = (2.04−2.07)/√(0.010²/30+0.015²/30) = −9.12. Critical value ±2.576 at α=0.01; |z|>2.576, so reject H0. The machines do not fill the same mean volume; machine B fills more (2.07 L vs 2.04 L).

## Answer

The population standard deviations are known (0.010 L and 0.015 L), so a two-sample z-test is used.

1. H0: μ_A = μ_B (both machines fill the same mean volume)
   H1: μ_A ≠ μ_B (two-tailed)
2. Level of significance: α = 0.01
3. Test statistic:

```text
z = (x̄_A − x̄_B) / √(σ_A²/n_A + σ_B²/n_B)
SE = √(0.010²/30 + 0.015²/30) = √(0.00000333 + 0.00000750) = 0.00329
z  = (2.04 − 2.07)/0.00329 = -9.115
```

4. Critical value: z(0.01, two-tail) = ±2.576. Reject H0 if |z| > 2.576.
5. Decision: |z| = 9.115 > 2.576 (p-value < 0.0001), so **reject H0**.

**Conclusion:** the two machines do not fill the same mean volume at the 1% level; machine B fills more (2.07 L against 2.04 L).

## Check yourself

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

1. Why is a z-test (not t) used for the two filling machines?
   A. Because the sample means are close
   B. Because the population standard deviations are known
   C. Because the samples are paired
   D. Because the variances are unequal
   Answer: B. Known sigma values (0.010 and 0.015 L) allow the z statistic with sqrt(sigma1²/n1 + sigma2²/n2). (Key point: Choice of test)

2. Which expression gives the standard error of x̄A − x̄B?
   A. sqrt(0.010² + 0.015²)
   B. (0.010 + 0.015)/sqrt(30)
   C. sqrt(0.010²/30 + 0.015²/30)
   D. 0.015/sqrt(30)
   Answer: C. Variances of the two independent means add, giving SE ≈ 0.00329 L. (Key point: Standard error)

3. What is the critical value for this two-tailed test at alpha = 0.01?
   A. ±1.645
   B. ±1.96
   C. ±2.326
   D. ±2.576
   Answer: D. alpha/2 = 0.005 in each tail gives z = ±2.576. (Key point: Critical value)

4. The calculated |z| is 9.115. What is concluded?
   A. Reject H0; the machines do not fill the same mean volume
   B. Fail to reject H0; the machines are the same
   C. Increase alpha and retest
   D. Only machine A is wrong
   Answer: A. 9.115 > 2.576, so H0 is rejected; machine B fills more (2.07 L vs 2.04 L). (Key point: Decision)

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