a large number of observations follow a normal distribution with a mean of 120 and a standard deviation of 80. What proportion of observations is:
i) below 100
ii) above 130
iii) between 100 and 130
iv) between 130 and 140
v) between 115 and 135??
anyone have an ideas because I certainly dont
Arrggghhh indeed !
Let me put my thinking cap on here ...
Well, you said "normal" distribution. That is a special word that says it follows a certain well-defined pattern on a graph that looks like a bell. (Funny that "normal" means it is special ...)
You can look at the curve here
First step is to figure how many standard deviations (80 in your case) that each value is from the mean (120 in your case).
Let us work on your examples:
i) below 100 ==> 100 is 20 away from the mean of 120, so it is 20/80 = 0.25 standard deviations away.
ii) above 130 ==> 130 is 10 away from the mean of 120, so it is 10/80 = 0.125 standard deviations away.
The next step is to look up these values (0.25 and 0.375), which are called "z-scores", in a table. Yep, you gotta look em up, unless you have them as a special function on your calculator or software.
Looking up these values, I get:
That means that 0.0987, or 9.87%, of the population are between the mean (120) and 100
And that 0.0478, or 4.78% are between the mean and 130
The rest is just figuring what the question asked: above, below, between?
The first question says "below 100" - well, we know that 9.87% are between 100 and 120, and 50% must be greater than 120 (because 120 is the mean), so 59.87% are GREATER, and therefore 41.2% must be LESSER, or below!
Likewise "above 130" can be figured out to be 45.2%
All the others can be worked out in a similar way, and I leave that up to you ... !
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman