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Worked solution

Use an exact normal distribution for plant mean height

The heights of a species of plants follow a normal distribution with mean 21 cm and standard deviation 90\sqrt{90} cm. A random sample of 10 plants is taken and the mean height calculated. Find the probability that this sample mean lies between 18 cm and 27 cm.

CJC · Lecture · Example 5

Normality is exact for any sample size

Let XX be one plant’s height in cm: X∼N(21,90)X\sim N(21,90). Under independent sampling, Xˉ∼N(21,90/10)=N(21,9)\bar X\sim N(21,90/10)=N(21,9) exactly. Its standard deviation is 3 cm, so P(18<Xˉ<27)=Φ(2)−Φ(−1)=0.8185946141≈0.819.\begin{aligned}&P(18<\bar X<27)\\ & =\Phi(2)-\Phi(-1)\\ & =0.8185946141\approx0.819.\end{aligned} No CLT is needed even though n=10n=10.

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Common mistakes

The radical on √90 is visible in both page images but lost in the plain text extraction. Treating 90 as the standard deviation would change the problem.

Solution notes

Teacher page 16 ends with the Example 5 heading; the actual question and working are on PDF 17 / printed 54. Source numerical model retained.

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