Porisma

Worked solution

Infer mean and standard deviation from two tail areas

Blue Neon Guppy masses at Green Park Tropical Fish Farm are normal with mean μ g and standard deviation σ g.5% exceed 85 g and 10% are below 25 g. Find μ and σ, then find the range symmetric about the mean containing 75% of the masses.

CJC · Lecture · Example 6

Use the upper-tail information

Let X be the mass in grams of a Blue Neon Guppy, X∼N(μ,σ2)X\sim N(\mu,\sigma^2). The Teacher fills the standardisation and first equation:

P(X>85)=0.05⟹P ⁣(Z>85−μσ)=0.05.P(X>85)=0.05\quad\Longrightarrow\quad P\!\left(Z>\frac{85-\mu}{\sigma}\right)=0.05.

Use invNorm(0.05,0,1,RIGHT) to get 1.64485362695… (the source shows 1.64485). Hence

85−μσ≈1.64485362695,μ+1.64485362695σ≈85.(1)\frac{85-\mu}{\sigma}\approx 1.64485362695,\qquad\mu+1.64485362695\sigma\approx 85.\qquad\text{(1)}

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Use the lower-tail information and solve

The second Teacher completion uses invNorm(0.1,0,1,LEFT):

P(X<25)=0.1⟹P ⁣(Z<25−μσ)=0.1.P(X<25)=0.1\quad\Longrightarrow\quad P\!\left(Z<\frac{25-\mu}{\sigma}\right)=0.1.

The lower quantile is −1.28155156554… (source display −1.281551567), giving

25−μσ≈−1.28155156554,μ−1.28155156554σ≈25.(2)\frac{25-\mu}{\sigma}\approx-1.28155156554,\qquad\mu-1.28155156554\sigma\approx 25.\qquad\text{(2)}

Subtract(2) from (1):

σ≈85−251.64485362695+1.28155156554=20.50297073,μ≈25+1.28155156554σ=51.27561423.\sigma\approx\frac{85-25}{1.64485362695+1.28155156554}=20.50297073,\qquad\mu\approx 25+1.28155156554\sigma=51.27561423.

Thus μ=51.3 g,σ=20.5 g\boxed{\mu=51.3\text{ g},\quad\sigma=20.5\text{ g}} (3 s.f.). The source retains the model N(51.27561426,20.502970722)N(51.27561426,20.50297072^2) for subsequent calculations. The Teacher specifically requires at least 5 significant figures in intermediate working; retaining full calculator values is preferable.

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Find the symmetric central range

The central area 0.75 leaves equal tails (1−0.75)/2=0.125(1-0.75)/2=0.125. Using the full model, invNorm(0.75,51.27561426,20.50297072,CENTER) gives endpoints approximately

a=27.6900346,b=74.8611939.a=27.6900346,\qquad b=74.8611939.

The source writes 27.690<X<74.86127.690<X<74.861. These unrounded endpoints define the central 75% region; both are equally distant from the mean. With ordinary nearest 3-significant-figure rounding, report approximately 27.7 g to 74.9 g\boxed{27.7\text{ g to }74.9\text{ g}}.

The Teacher also says to explain rounding involving inequalities. Its next line prints 27.7 g and 74.8 g, calling both 3 s.f. The upper 74.8 is not the ordinary nearest 3 s.f. rounding of 74.861…. It can be read as an inward endpoint approximation; that inward interval contains approximately 0.7492845 rather than exactly 0.75. Keep the full bounds when an exact central probability or symmetry matters, and do not claim the rounded bounds retain exactly 75%.

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Three regions for guppy mass in grams. Left of 25 has area 0.10, right of 85 has area 0.05. Solving their standard-normal quantile equations gives mean 51.27561423 and standard deviation 20.50297073. Central area 0.75 has endpoints 27.69003456 and 74.86119391, symmetric about the mean. The source-reported 74.8 is an inward approximation, not ordinary nearest three-significant-figure rounding.
Three regions for guppy mass in grams. Left of 25 has area 0.10, right of 85 has area 0.05. Solving their standard-normal quantile equations gives mean 51.27561423 and standard deviation 20.50297073. Central area 0.75 has endpoints 27.69003456 and 74.86119391, symmetric about the mean. The source-reported 74.8 is an inward approximation, not ordinary nearest three-significant-figure rounding.
Three regions for guppy mass in grams. Left of 25 has area 0.10, right of 85 has area 0.05. Solving their standard-normal quantile equations gives mean 51.27561423 and standard deviation 20.50297073. Central area 0.75 has endpoints 27.69003456 and 74.86119391, symmetric about the mean. The source-reported 74.8 is an inward approximation, not ordinary nearest three-significant-figure rounding.

Common mistakes

Keep at least 5 s.f. in intermediate working, then distinguish μ/σ from the central interval endpoints. Source upper endpoint 74.8 is not ordinary nearest 3 s.f. rounding of 74.861…; use the unrounded bounds to define the exact central 75%.

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