SOLUTION: Please solve this equation: The mean of a normal probability distribution is 300; the standard deviation is 16. a. About 68% of the observations lie between what two values?

Algebra ->  Statistics  -> Normal-probability -> SOLUTION: Please solve this equation: The mean of a normal probability distribution is 300; the standard deviation is 16. a. About 68% of the observations lie between what two values?       Log On


   

Question 1201663: Please solve this equation:
The mean of a normal probability distribution is 300; the standard deviation is 16.
a. About 68% of the observations lie between what two values?
______________

b. About 95% of the observations lie between what two values?
___________________________
c. Practically all of the observations lie between what two values?
_____________________



Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answers:
  1. 284 and 316
  2. 268 and 332
  3. 252 and 348

Explanation:

We use the Empirical Rule.
Some textbooks call it the 68-95-99.7 rule but I don't think it's as catchy of a name.

The rule has 3 properties:
  • roughly 68% of the normal distribution is within 1 standard deviation of the mean.
  • roughly 95% of the normal distribution is within 2 standard deviations of the mean.
  • roughly 99.7% of the normal distribution is within 3 standard deviations of the mean.



For part (a), the work shown would be something like this
mu - 1*sigma = 300 - 1*16 = 284
mu + 1*sigma = 300 + 1*16 = 316
Roughly 68% of the observations are between 284 and 316.

For part (b), we have
mu - 2*sigma = 300 - 2*16 = 268
mu + 2*sigma = 300 + 2*16 = 332
Roughly 95% of the observations are between 268 and 332.

For part (c), the phrasing "practically all" means "practically 100%" which is close enough to 99.7%
mu - 3*sigma = 300 - 3*16 = 252
mu + 3*sigma = 300 + 3*16 = 348
Roughly 99.7%, aka practically all, of the observations are between 252 and 348.