Question

Let X be normally distributed with mean and variance.
a) Verify that q3 = mean + 0.67σ and that q1 = mean - 0.67σ.
b) Find the interquartile range for X.
c) Verify that the inner fences for X are f1 = mean - 2.68σ and f3 = mean + 2.68σ.
d) Verify that the probability that X will fall beyond the inner fences is approximately 0.007.

Answer 1

Okay so starting with (A):
Q3 and Q1 are interquartile ranges, so Q1 would have a probability of 0.25, and Q3 would have a probability of 0.75. Right?
Now that we have the probability for both of them, we can find their z-scores by doing Inverse-Normal on a calculator (TI-Nspire), or however way your teacher taught you.

You can substitute 'mean' as 0, and 'σ' as 1 to standardize the values, since a z-score is standardized.

(B) and (C) looks like they require a graph or diagram of some sort. Do you need to screenshot it?

Answer 2

Yeah, ok that makes sense. Why is the mean 0? There's no visual for it. It just says X is normally distributed with mean μ and variance σ.

Answer 3

any clue @mhchen ?

Answer 4

Yeah, it's just giving you the variables.
Here is the standard normal curve, which is used to find z-scores:


As you can see, the x-axis of -2,-1,0,1,2,3 are z-scores.
The area UNDERNEATH the curve is the probability, given that is happens before a z-score