## Central Limit Theorem: “Between” Probability

Sample problem: The there are 250 dogs at a dog show who weigh an average of 12 pounds, with a standard deviation of 8 pounds. If 4 dogs are chosen at random, what is the probability they have an average weight of greater than 8 pounds and less than 25 pounds?

Step 1:Identify the parts of the problem. Your question should state:

• mean (average or μ) standard deviation (σ) population size
• sample size (n)
• number associated with “less than” 1
• number associated with “greater than” 2

Step 2: Draw a graph. Label the center with the mean. Shade the area between   1 and   2. This step is optional, but it may help you see what you are looking for.

Step 3: Use the following formula to find the z-values.

All this formula is asking you to do is:

a) Subtract the mean (μ in Step 1) from the greater than value (Xbar in Step 1): 25-12=13.
b) Divide the standard deviation (σ in Step 1) by the square root of your sample (n in Step 1): 8/sqrt4=4
c) Divide your result from a by your result from b: 13/4=3.25

Step 4 Use the formula from Step 3 to find the z-values. This time, use Xbar2 from Step 1 (8).

a) Subtract the mean (μ in Step 1) from the greater than value (Xbar in Step 1): 8-12=-4.
b) Divide the standard deviation (σ in Step 1) by the square root of your sample (n in Step 1): 8/sqrt4=4
c) Divide your result from a by your result from b: -4/4= -1

Step 5: Look up the z-value you calculated in Step 3 in the z-table.

Z value of 3.25 corresponds to .4994

Step 6: Look up the z-value you calculated in Step 4 in the z-table.

Z value of 1 corresponds to .3413

Step 7: Add Step 5 and 6 together:

.4994 + .3413 = .8407

Step 8: Convert the decimal in Step 7 to a percentage:

.8407 = 84.07%

## 4 thoughts on “Central Limit Theorem: “Between” Probability”

1. Ivonne

I followed you till we got to the Z-Chart
My charts (looked it up in two different sources of Z-Charts) gave me .9994 for 3.25 and .1587 for -1. Another question, why did you change the -1 to a 1. I still would have a different number for a 1.

2. Andale Post author

Ivonne,

There are several z-tables. The one I used is from the Elementary stats book we used at Florida State College in Jax. It’s also hyperlinked in the actual article.

As the z-table is about areas, it doesn’t make sense to have a negative area, so that’s why -1 became 1.

Hope that helps :)

Stephanie