Porisma

Worked solution

Find a central eighty-percent interval

If X∼N(70,25)X\sim N(70,25), find aa such that P(∣X−70∣<a)=0.8P(|X-70|<a)=0.8. Hence find the limits containing the central 80% of the distribution.

CJC · Lecture · Example 3

Translate the central area

The mean is 70 and σ=5. Expand the modulus:

P(∣X−70∣<a)=P(−a<X−70<a)=P(70−a<X<70+a)=0.8.\begin{aligned}&P(|X-70|<a)\\ & =P(-a<X-70<a)\\ & =P(70-a<X<70+a)\\ & =0.8.\end{aligned}

The bounds are equally distant from μ70. The remaining probability 0.2 splits equally into tails of 0.1.

How did this part go?

Use either returned limit to find a

In the source’s invNorm wizard, enter area 0.8, μ70 and σ5, select CENTER, then Paste and ENTER. invNorm(0.8,70,5,CENTER) returns the pair

{63.59224217, 76.40775783}.\{63.59224217,\ 76.40775783\}.

Either limit gives the same half-width. Using the lower limit,

70−a≈63.59224217⟹a≈6.40775783.70-a\approx 63.59224217\quad\Longrightarrow\quad a\approx 6.40775783.

Using the upper limit,

70+a≈76.40775783⟹a≈76.40775783−70=6.40775783.70+a\approx 76.40775783\quad\Longrightarrow\quad a\approx 76.40775783-70=6.40775783.

Thus a=6.41\boxed{a=6.41} (3 s.f.). Keep the full calculator values until the final rounding. If CENTER is unavailable, use LEFT areas 0.1 and 0.9 to obtain the same two limits.

How did this part go?

State the interval limits

The central 80% lies between 63.6 and 76.4\boxed{63.6\text{ and }76.4} (3 s.f.). CENTER returns the two endpoints, not the half-width a; subtract the mean or subtract the lower endpoint from the mean to find a.

How did this part go?

Compare your sketch

Expand diagram

Use the scrollbars or arrow keys to inspect details.

Normal mean 70 variance 25. Shaded central area 0.8 lies from 63.59224 to 76.40776. Equal unshaded tails have area 0.1 each. The half-width is a≈6.40776, so the reported limits are 63.6 and 76.4 to three significant figures.
Normal mean 70 variance 25. Shaded central area 0.8 lies from 63.59224 to 76.40776. Equal unshaded tails have area 0.1 each. The half-width is a≈6.40776, so the reported limits are 63.6 and 76.4 to three significant figures.
Normal mean 70 variance 25. Shaded central area 0.8 lies from 63.59224 to 76.40776. Equal unshaded tails have area 0.1 each. The half-width is a≈6.40776, so the reported limits are 63.6 and 76.4 to three significant figures.

Common mistakes

CENTER applies only when the desired bounds are symmetric about the mean. Do not mistake an endpoint 76.4078 for the half-width 6.4078. Use σ=√25=5.

Read the hint