Answer to Question #295866 in Statistics and Probability for Tom

Question

Compute a range of yearly clothing expenditures—

measured in dollars—that includes 80% of all students

on this campus? Explain why any number

of such ranges could be found, and find the

shortest one.

Answer 1

Question is incomplete.

Full question is:

It is known that amounts of money spent on clothing in a year by students on a particular campus follow a normal distribution with a mean of $380 and a standard deviation of $50. Compute a range of yearly clothing expenditures— measured in dollars—that includes 80% of all students on this campus? Explain why any number of such ranges could be found, and find the shortest one.

Solution:

There are an infinite number of pairs of values a and b such that P(a<Z<b)=0.8 .

The shape of the bell curve causes the distance between a and b to be minimized if we center this interval on zero (which means a=-b ).

Therefore:

$0.8=P\left(z_{-b}<Z<z_{b}\right)$

And $0.4=P\left(Z<z_{b}\right)$

Therefore $z_{b} \cong 1.28$

$\begin{aligned} (a, b) & \cong(\mu-1.28 \sigma, \mu+1.28 \sigma) \\ & \cong(380-1.28(50), 380+1.28(50)) \\ & \cong(316,444) \end{aligned}$