I have a normal probability distribution question which does not have the standard deviation provided. Help!?

Question

Use the sales forecaster's prediction to describe a normal probability distribution that can be used to approximate the demand distribution. Sketch the distribution and how its mean and standard deviation.

Provided information: Sales forecaser predicts an expected demand of 20,000 units with a 0.90 probability that demand would be between 10,000 units and 30,000 units.

What is the standard deviation?

Answer 1

We are given the mean is 20,000?

So P(10k < X < 30k) = 0.9
by symmetry
P(20k < X < 30k) = 0.45
P(0 < Z < 10k/σ) = 0.45

From normal tables, z value corresponding to this 0.45
= 1.645
So 10000/σ = 1.645
σ = 6079

Answer 2

You know that 90% of the normal distribution falls between 10000 and 30000, so 45% falls between 20000 and 30000. Looking up on a normal distribution chart tells me that 45% of the weight means 1.645 standard deviations (roughly). The standard deviation is thus 6080, which I would round to 6000.

You should also look through your textbook or notes to make sure there isn't a different method for finding the confidence interval of a single measurement like this.

Answer 3

The manufacturer wants to report a number x such that 90% of cartridges print at least x pages. Typically, a z-table gives the probability that a normally-distributed variable is less than z. You want the z such that 90% of outcomes are *more* than z, so look up the probability 10% (as a decimal, 0.1) Find 0.1 in the list of probabilities and read off the z-value associated with it. z = -1.28 Now convert from z to x -1.28 = (x - 12450)/570 x = 11720

Answer 4

note
I am going to make some assumptions here:
Since the mean is the best guess, and they are forecasting a sale of 20,000 units
x-bar = 20,000
With a 90% probability, then alpha = .10 and alpha/2 = .05
so using .0500 as the right tail, z = 1.645 which is associated with x = 30,000

Now,

z = (x - xbar)/sd
1.645 = (30,000 - 20,000)/sd
1.645 = 10,000/ sd
sd = 10,000/1.645 = 6,079.03

assuming my assumptions are assumable

Answer 5

If the standard deviation is σ, then we have
P(-10000/σ < Z < 10000/σ) = 0.90
where Z is a standard normal variable.
Since this is symmetrical, we know
P(Z < 10000/σ) = 0.95
and hence 10000 / σ = 1.645 (4 s.f.)
so σ= 6080 to 3 s.f.

I trust you can handle the rest from here.

Answer 6

The formula is square root of (b-a) squared/12.

Another formula is 2* sqrt 3 - mean

Both should get you the same answer if computed properly.