r a particular value, this table gives the percent of scores between the mean and the z-value of a normally distributed random variable. What percent of the total population is found between the mean and the z-score, assume z = 2.57.
To determine the percent of the total population between the mean and a specific z-score, we can use the standard normal distribution table, also known as the Z-table.
The Z-table provides the percentage of scores that fall between the mean (0) and a given z-score. Each entry in the table represents the percentage of scores that fall below a specific z-score.
However, most Z-tables don't provide values beyond a certain z-score. Normally, a Z-table only goes up to a z-score of 3.49. In your case, you mention that z = 2.57, which falls within the range of most standard Z-tables.
To find the percentage of the total population between the mean and z = 2.57, follow these steps:
1. Locate the row corresponding to the whole number of the z-score in the table. In this case, find the row for z = 2.5.
2. Locate the column corresponding to the second decimal place of the z-score. In this case, look for the column for z = 0.07 (since 2.57 - 2.50 = 0.07).
3. The intersection of the row and column in the table will give you the percentage of scores that fall below that z-score. In this case, the value should be 0.9951.
4. Subtract this value from 0.5000 (the percentage of scores that fall below the mean), which gives you the percentage of scores that fall between the mean and z = 2.57.
0.5000 - 0.9951 = -0.4951
Note: The resulting value may appear negative, but it simply indicates the area under the curve beyond the given z-score. To find the positive value, you need to take the absolute value (ignore the negative sign).
Absolute value of -0.4951 = 0.4951
Therefore, approximately 49.51% of the total population lies between the mean and z = 2.57.