Home / IB Mathematics AHL 4.16 Confidence intervals for the mean of a normal population.AI HL Paper 1- Exam Style Questions

IB Mathematics AHL 4.16 Confidence intervals for the mean of a normal population. AI HL Paper 1- Exam Style Questions- New Syllabus

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Question

An ecologist records the water temperatures at \(5\) distinct sites across a coral reef. The sample yields a mean temperature of \(20.1^{\circ}\text{C}\) and a sample standard deviation, \(S_n\), of \(3.2^{\circ}\text{C}\).
(a) Calculate an unbiased estimate for the variance of the entire reef’s population.
(b) Given the assumption that reef temperatures follow a normal distribution, determine a \(95\%\) confidence interval for the population mean temperature.
(c) Based on your confidence interval from part (b), evaluate whether it is reasonable to suggest that the true mean temperature of the reef could be \(17^{\circ}\text{C}\).

Most-appropriate topic codes:

• AHL 4.14: Unbiased estimate of population variance — part (a)
• AHL 4.16: Confidence intervals for the mean — part (b)
▶️ Answer/Explanation
Detailed solution

(a)
The unbiased estimate of the population variance, \(s_{n-1}^2\), is given by:
\(s_{n-1}^2 = \frac{n}{n-1} S_n^2\)
\(s_{n-1}^2 = \frac{5}{4} \times (3.2)^2\)
\(s_{n-1}^2 = 1.25 \times 10.24 = 12.8\)

(b)
Since the population standard deviation is unknown and the sample size is small (\(n=5\)), use the t-distribution.
Using a GDC (TInterval):
\(\bar{x} = 20.1\)
\(s_x = \sqrt{12.8} \approx 3.5777\)
\(n = 5\)
Confidence Level \(= 0.95\)

Interval: \([16.1, 24.1]\) (to 3 s.f.)

(c)
The value \(17\) lies within the confidence interval \([16.1, 24.1]\). Therefore, it is plausible that the mean temperature is \(17^{\circ}\text{C}\).

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