Help with Stats

Question

The population of weights of a particular fruit is normally distributed, with a mean of 432 grams and a standard deviation of 30 grams. If 4 fruits are picked at random, then 14% of the time, their mean weight will be greater than how many grams?

Answer 1

HI Jordi,

This problem involves the classic equation in elementary statistics z=(x-mu)/sigma, but with a twist. We are working with a sample mean here, which means we need a special standard deviation metric for samples to replace sigma. This is called the standard error, and it has its own formula:

SE=sigma/sqrt(n)

where sigma=known population standard deviation

n=sample size

So, here:

sigma=30

n=4

SE=30/sqrt(4)

SE=15

Now, we replace sigma in the initial classic with the standard error. We now have:

z=(x-mu)/SE Breaking this down:

x=value we seek

mu=432

z=z-critical value explained below

Now, we were asked "14% of the time, mean weight will exceed how many grams?" I initially thought we were looking for the 14th percentile, but, since percentiles go bottom to top, we are actually looking for the 86th. That's the only way we can get the greater than value. So, go to the z-table, search the interior for 0.86, and record the corresponding value. The closest I found was 0.8599 where:

z=1.08

Substituting into our classic:

1.08=(x-432)/15

16.2=x-432

x=448.2 grams

I hope this helps.