what percebt of total population is found betweenthe mean and the z-score, assume z=2.57
mean --> z score of 0
my method of choice for these questions:
http://davidmlane.com/normal.html
click on "between" , enter 0 and 2.57 to get
.4949 or 49.49%
To find the percentage of the total population between the mean and a specific z-score, you need to use a standard normal distribution table or a calculator with a built-in function for calculating normal probabilities.
Here's how you can find the percentage:
1. Determine the area to the left of the z-score: The area to the left of the z-score is the cumulative probability up to that point. In this case, since you want the percentage between the mean and the z-score, you are interested in the area to the left of the z-score.
2. Look up the cumulative probability in the standard normal distribution table or use a calculator: You can find standard normal distribution tables online, or many scientific calculators have built-in functions to calculate normal probabilities. For a z-score of 2.57, you want to find the cumulative probability to the left of 2.57.
3. Calculate the percentage: The cumulative probability gives you the area to the left of the z-score. To find the area between the mean and the z-score, subtract the cumulative probability from 0.5 (which represents 50% of the total area under the curve because the normal distribution is symmetrical).
For example, if the cumulative probability up to 2.57 is 0.9951, then the percentage of the total population between the mean and the z-score is:
Percentage = 0.5 - 0.9951 = -0.4951
Since the result is negative, it means that the z-score is beyond 50% of the total population. In this case, it is beyond the right tail of the distribution.
Remember to always check the direction of the z-score (positive or negative) when using a standard normal distribution table or calculator.