Statistics: Solve for z and P.

[Frustrated01]

I'm so happy I found this forum.

I'm not asking for anyone to complete my work, but merely provide a bit of insight to completing same. I missed the class due to illness.

What is the probability?
z is less than or equal to -1.5 z=
z is less than or equal to +1.5 z=
z is greater than of equal to 1.5 z=
z is greater than of equal to -1.5 z=
z is greater than 0 and less than 2.0 z=
z is greater than -1.5 and less than 0 z=

What is z?
the area to the left of z is .975 P=
the area to the left of z is .025 P=
the area between 0 and z is .475 P=
the area between 0 and z is -.475 P=
the area between z and -z is .246 P=
the area between z and -z is .5 P=

[jemidiah]

I assume the probability density function in question is the standard normal PDF (as discussed here). Given a probability density function f(z), the probability of z being between a and b is just the area under f(z) for z between a and b. For instance, suppose the standard normal PDF describes the amount of error in a measurement of your height. Say z is in inches, so z=0" means the measurement is perfect, z=-1" means the measurement was too low by 1", z=12" means the measurement was too high by a foot, etc. Clearly z=about 12" is *much* less likely than z=about 1", unless you're horrifically bad at measuring heights. So, the normal PDF is actually decent for describing this situation, since it decreases very rapidly, forcing error measurements far from 0 to be highly unlikely.

Given the above situation, what's the chance that your measurement is too high? Phrased, more technically, what's the probability that z is between 0 and +infinity? By definition, this is the area between the f(z) curve and the x-axis for z between 0 and +infinity. PDFs by definition have total area 1 (area between -infinity and +infinity), and the standard normal PDF is symmetric about z=0. So, the total area between -infinity and 0 of f(z) is 1/2, as is the total area between 0 and +infinity. So, the probability the measurement is too high is 1/2. Note, interestingly, that the probability that the measurement is too low is also 1/2, forcing the probability that the measurement is exactly right to 0. This actually makes sense: out of all the numbers you might get, only one of them is *exactly* right, while an infinite number of them are at least very *slightly* off.

The first question, probability z <= -1.5, is very similar. It requires you to calculate the area between f(z) and the x-axis for z between -infinity and -1.5 To to this requires the use of a table (or a numeric integration system, though I dunno if you've had calculus). The table basically just contains the areas between -infinity and a specific point, say x. There is a name for the area between -inf and a point, x, of a probability density function: the cumulative distribution function. In this case, you would need to look up the value of the cumulative distribution function of the standard normal distribution for x=-1.5.

The table on the second page of this pdf lists a slightly modified cumulative distribution function. For instance, the area between -inf and 1.02 is 0.8461, i.e. an ~85&#37; chance. You need to be slightly tricky to use the table for the first question you were given, the probability that z <= -1.5. From the symmetry of the pdf, the area under z between -1.5 and 0 is the same as that between 0 and 1.5. Also, the area between -inf to 0 is 1/2 as reasoned above. So, we have the area between -inf and 1.5 from the table as 0.9332, showing the area between 0 and 1.5 is just the area between -inf and 1.5 minus the area between -inf and 0, i.e. 0.9332 - 0.5 = 0.4332. Thus the area between -1.5 and 0 is also 0.4332. Now, what's the area between -inf and -1.5? Simply the area between -inf and 0 minus the area between -1.5 and 0, or 0.5-0.4332 = 0.0668. That is, you have only a ~7% chance of have z less than -1.5. This is the answer to the first question: p=0.0668.

As for the "what is z" questions, they're basically the same as the above in reverse. For instance, the area left of z is 0.975 is the value of z for which the CDF is 0.975. From the table linked, this happens at z=1.96.

I'll of course leave the other variants unanswered.