# [SOLVED]is normally distributed

#### karush

##### Well-known member
The lifespan of a particular species of insect is normally distributed with a mean of $57$ hours and a standard deviation of $4.4$ hours.

$90\%$ of the insects die after $t$ hours.

Represent this information on a standard normal curve diagram,
indicating clearly the area representing $90\%$.
$.9$ on the z-table $\displaystyle \approx z=1.29$. so from W|A

Find the value of t.
$\displaystyle\frac{t-57}{4.4}=1.29$ thus $t\approx 63s$

any correction #### Klaas van Aarsen

##### MHB Seeker
Staff member
Hi karush.

I find the problem statement a bit ambiguous: "$90\%$ of the insects die after $t$ hours."
I guess you interpreted it as "$90\%$ of the insects are dead after $t$ hours." or "$90\%$ of the insects die after $t$ hours or less."
I'm wondering if they intended: "$90\%$ of the insects die after at least $t$ hours."
In that case you'd get a different answer, but I'm not sure if that's what was intended.
Perhaps you can clarify.

Anyway, for the rest your calculation is fine.

One correction: I don't think they die after 63 seconds which would be a ridiculously short lifespan. Shall we make it hours?

#### MarkFL

##### Administrator
Staff member
My only contribution is according to my table the $z$-score is closer to 1.28, and the computer gives an approximate value of 1.28155.

#### karush

##### Well-known member
ok, i got the bugs out of it.

$\displaystyle\frac{t-57}{4.4}=1.28155$

$t = 62.6432 or \approx 63h$

using the wiki z-table which only goes to hundredths; is there a W|A calculator for getting more digits?

yes i wasn't sure about what they were asking? maybe it should look this

View attachment 1150

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#### MarkFL

##### Administrator
Staff member
I used the command in W|A:

erf(z/sqrt(2))=2(0.4)

and it returns:

$$\displaystyle z\approx1.28155$$

#### Klaas van Aarsen

##### MHB Seeker
Staff member
using the wiki z-table which only goes to hundredths; is there a W|A calculator for getting more digits?
I have given up on W|A for statistical calculations.
Instead I recommend Excel.
Type in a cell "=NORMSINV(0.9)" and you'll get your result $1.281551566$.

#### karush

##### Well-known member
well that is good to know. Its hard to use a table, often choosing the one really not the closest.

thank again. gems of know how every time...