---
title: "Test: mean weight of ten objects is 65 g"
subject: "Probability and Statistics"
chapter: 6
key: "semester_5/probability_and_statistics:weights-ten-objects-mean-65"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/weights-ten-objects-mean-65
site: BCT Engineering
---

# Test: mean weight of ten objects is 65 g

Probability and Statistics · Chapter 6: Inference concerning mean

- Asked in: 2072 Chaitra, 5 marks
- Video: [Hypothesis Testing for Population Mean ,Problem 1| 21MAT41-JOINT Probability& Sampling theory#vtu](https://www.youtube.com/watch?v=NT7JpggJUBk) — VTU padhai
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/weights-ten-objects-mean-65
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6

## Question

Ten objects were chosen at random from the large population and their weights were found to be in grams 63, 63, 64, 65, 66, 69, 65, 66.1, 64.5. In the light of above data, discuss the suggestion that the mean weight in the population is 65 gm. Use α = 0.05.

## Short answer

One-sample t-test: n = 9, mean = 65.07 g, t = 0.108, df = 8. At α = 0.05, critical value = ±2.306; fail to reject H0. The data support the suggestion that the population mean weight is 65 g.

## Answer

Population SD is unknown and the sample is small, so a one-sample t-test is used. (Only nine weights are listed although the question says ten objects; the nine given values are used, n = 9.)

1. H0: μ = 65 g
   H1: μ ≠ 65 g (two-tailed)
2. Level of significance: α = 0.05
3. Sample values and test statistic:

```text
n = 9, Σx = 585.6, x̄ = 65.0667
s² = Σ(x−x̄)²/(n−1) = 27.4200/8 = 3.4275,  s = 1.8514
t = (x̄ − μ₀)/(s/√n) = (65.0667 − 65)/(1.8514/√9) = 0.0667/0.6171 = 0.108
```

4. Critical value: t at α = 0.05, df = n−1 = 8: ±2.306. Reject H0 if |t| > 2.306.
5. Decision: the calculated value falls in the acceptance region (p-value = 0.9166), so **fail to reject H0**.

**Conclusion:** the data are consistent with the suggestion that the population mean weight is 65 g.

## Check yourself

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

1. Which test is used?
   A. ANOVA
   B. z-test
   C. One-sample t-test, sigma unknown and n small
   D. Paired t
   Answer: C. A single small sample tested against mu0 = 65 with unknown sigma. (Key point: Choice of test)

2. How many values are used and why?
   A. Nine, because only nine weights are listed
   B. Eight, dropping extremes
   C. Eleven
   D. Ten, as the question says
   Answer: A. Only nine values are given, so n = 9 and df = 8. (Key point: Data note)

3. How is t computed?
   A. xbar/s
   B. (xbar - 65)/s
   C. (xbar - 65) x sqrt(n)
   D. (xbar - 65)/(s/sqrt(n))
   Answer: D. t = 0.0667/0.6171 = 0.108. (Key point: Test statistic)

4. What is the decision?
   A. t < 0, reject
   B. |t| > 2.306, reject
   C. |t| = 0.108 < 2.306, fail to reject H0
   D. Reject because mean is 65.07
   Answer: C. The data are consistent with a mean of 65 g. (Key point: Decision)

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