---
title: "CI: sample of 16, mean 52, SD 4"
subject: "Probability and Statistics"
chapter: 6
key: "semester_5/probability_and_statistics:ci-sample-16-mean-52-sd"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/ci-sample-16-mean-52-sd
site: BCT Engineering
---

# CI: sample of 16, mean 52, SD 4

Probability and Statistics · Chapter 6: Inference concerning mean

- Asked in: 2069 Chaitra, 4 marks
- Video: [Introduction to Confidence Intervals for the Mean.mp4](https://www.youtube.com/watch?v=oiinmoA3t0Y) — Dr. Harvi Millar
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/ci-sample-16-mean-52-sd
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6

## Question

A random sample of size 16 showed a mean of 52 with a standard deviation 4. Obtain 99% and 95% confidence limits population mean.

## Short answer

Small sample (n=16) with unknown σ, use t-interval: 99% CI = 52 ± 2.947*(4/√16) = (49.05, 54.95); 95% CI = 52 ± 2.131*1 = (49.87, 54.13).

## Answer

n = 16 is small and σ is unknown, so the t-interval x̄ ± t(α/2, n−1)·s/√n with 15 d.f. is used.

```text
n = 16, x̄ = 52, s = 4, df = 15
SE = s/√n = 4/√16 = 1.0000

99% CI: t(0.005, 15) = 2.947
  margin = 2.947 × 1.0000 = 2.9467
  limits = 52 ± 2.9467 = (49.053, 54.947)

95% CI: t(0.025, 15) = 2.131
  margin = 2.131 × 1.0000 = 2.1314
  limits = 52 ± 2.1314 = (49.869, 54.131)
```

**Answer: 99% limits: (49.05, 54.95); 95% limits: (49.87, 54.13).**

## Check yourself

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

1. Which distribution and df?
   A. t with 15 d.f.
   B. z
   C. t with 16 d.f.
   D. chi-square
   Answer: A. n = 16 small, sigma unknown, df = 15. (Key point: Distribution)

2. What is SE?
   A. 4/sqrt(16) = 1
   B. 16/4 = 4
   C. 4 x 4
   D. 4/16
   Answer: A. SE = 1. (Key point: Standard error)

3. Which t values for 99% and 95%?
   A. 2.576 and 1.96
   B. 2.602 and 1.753
   C. 2.947 and 2.131
   D. 3.250 and 2.262
   Answer: C. With df = 15. (Key point: Critical values)

4. What are the limits?
   A. 99%: (50, 54)
   B. 99%: (49.05, 54.95); 95%: (49.87, 54.13)
   C. Both (48, 56)
   D. 99%: (49.87, 54.13); 95%: (49.05, 54.95)
   Answer: B. Higher confidence gives wider limits. (Key point: Intervals)

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