central limit theorem

Question

An unknown distribution has the following parameters: µx = 45 and σ x = 8 a sample size of 50 is drawn randomly from the population. Find the 80th percentile for the sum of the 50 values of x

Answer 1

So there are a few different versions of the Central Limit Theorem (CLT), depending on what you're trying to describe the distribution of (i.e., sum, average, standardized mean, etc.). They all ultimately say the same thing. Since we're interested in a sum, we're going to use this version:

∑x ~ N(mean = nµ_x, sd = σ_x√n).

Now for the specifics of the problem:

"µ_x = 45 and σ_x = 8 a sample size of 50 is drawn randomly from the population."

Now we can specify a distribution:

∑x ~ N(mean = nµ_x, sd = σ_x√n)

∑x ~ N(mean = (50)(45), sd = (8)√(50))

∑x ~ N(mean = 2250, sd = (8)√(50))

"Find the 80th percentile for the sum of the 50 values of x."

I'm assuming you can have a calculator for this. For a percentile, you're needing to find the value on this distribution we just found that has 80% of data below it, and therefore the remaining 20% above it. Since you know these percentages already (and really, they're probabilities), and since you need the raw values, you need an inverse procedure that works on the normal distribution. So invNorm!

On the TI-84:

2nd VARS (to get DISTR) ---> 3: invNorm(

A dialog box appears. Put 0.8 for area (since the area goes from the left to the right), 2250 for µ, and 8√(50) for σ. Press Paste and ENTER. The calculator outputs 2297.609287.

The 80th percentile for the sum of the 50 values of x is 2297.61.