a sample size 85 will be drawn from a population with mean 22 and standard deviation 8. find the probability that x will be between 21 and 24

Question

a sample size 85 will be drawn from a population with mean 22 and standard deviation 8. find the probability that x will be between 21 and 24

Answer 1

Ok, so we are dealing with a sample size of 85 drawn from a population with mean, μ=22, and standard deviation,σ=8.  Since we are dealing with a sample-- and not a population--we will create a distribution of sample means, for which:

 

(Please note: the notations I am using for sample might be different than the ones you were taught, but the formulas will remain the same)

 

   POPULATION                  vs.                          SAMPLE

mean= μ                                                        mean of sample M (M=μ) (this also can be denoted as μ_m instead of M)

standard dev= σ                                             standard dev = σ_m (σ_m = σ/√n) (this is also known as standard error)

N=size  (assumed infinite)                                n=size (number in sample)

 

Therefore the mean of our sample will be M=22

The n of our sample will be n=85

And the standard deviation of our sample will be σ_m = σ/√n  -->  σ_m = 8/√85  --> σ_m = 0.8677

 

Next we will use a z score transformation. Here is the formula to convert raw scores to z scores:

 

z= (x-M)/σ_m

 

 

x=raw score

M=mean

σ_m=standard deviation of sample (also = σ/√n)

 

 

To find the probability that x will be between 21 and 24, we will calculate their respective z scores, then use a z table to look up their probabilities.

 

x=21

z=(21-22)/0.8677

z= -1.15

 

x=24

z=(24-22)/0.8677

z= 2.30

 

(remember the z score for the mean is always transformed into 0)

 

Now it's time to go to a table of z scores (this should be in the appendix of your stats book, otherwise you can look them up online using this link http://faculty.kutztown.edu/rryan/classes/stats/resources/z_table.pdf)

 

There are a couple different ways to do this. I prefer using the D column or the proportion between Mean and z. If you have any questions about how to do it other ways you can respond below.

 

For z=-1.15 (remember the normal distribution is symmetrical, so you will use z=+1.15 to find the percentage) the proportion between mean and z=0.3749

 

For z=2.30, the proportion between mean and z=0.4893

 

Therefore the probability that the sample will be between 21 and 24 is 0.3749 + 0.4893= 0.8642, or 86.42%.