
Aug 21st, 2010, 07:44 AM
#1
Thread Starter
New Member
Statistics: Solve for z and P.
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=

Aug 22nd, 2010, 12:42 AM
#2
Re: Statistics: Solve for z and P.
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 xaxis 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 xaxis 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% 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 0.50.4332 = 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.
The time you enjoy wasting is not wasted time.
Bertrand Russell
< Remember to rate posts you find helpful.
Posting Permissions
 You may not post new threads
 You may not post replies
 You may not post attachments
 You may not edit your posts

Forum Rules

Click Here to Expand Forum to Full Width
