How do I find the range of lengths in this question?

Question

The length of a Daschund is normally distributed with a mean of 15 inches, and a standard deviation of 3.5 inches. What range of lengths would occur 75% of the time?

My main issue is that I don't know how to find the z scores in the context of this question.

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Answer 1 · 2018-06-26T09:51:07

The range would be #15+-4.025#, ie #(10.975, 19.025)#

#75%# of the time is #0.75/2 = 0.375#

Looking at the standard normal table below, this corresponds to the #Z# value of #1.15# (the exact area under the curve in the table is #0.3749#).

Therefore,

#Z = (X - "mean")/(SD) # = #(X - 15)/(3.5)#

Solving for #X#

#X = "mean" +- 4.025#

#= 15 +- 4.025#.