A statistics teacher believes that the final exam grade for her elementary stats class have a normal distribution with a mean of 82 and a standard deviation on 8.

Find the score which separates the top 10% of the scores from the lowest 90% of the scores.

z-score for 90

= (90-82)/8 = 1

so don't even have to go to tables or charts, we are 1 standard deviation from 82, so
a score of 90 will do it

To find the score which separates the top 10% of the scores from the lowest 90% of the scores, we need to find the value that corresponds to the 90th percentile.

To do this, we will make use of the standard normal distribution table. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. By standardizing the original distribution, we can find the corresponding z-score for the 90th percentile.

The formula to compute the z-score is:

z = (x - μ) / σ

where:
- x is the value we want to find
- μ is the mean of the distribution (given as 82)
- σ is the standard deviation of the distribution (given as 8)
- z is the z-score that corresponds to the desired percentile, which we want to calculate

To find the z-score corresponding to the 90th percentile, we need to find the z-score that leaves 10% in the tail of the distribution. Since the normal distribution is symmetric, we can find this z-score using the standard normal distribution table.

The standard normal distribution table provides the area under the curve to the left of a given z-score. Since we want the area to the right of the z-score (tail), we will subtract the area from 1.

Looking up the z-value for a cumulative probability of 0.90 in the standard normal distribution table, we find that it is approximately 1.28.

Now we can substitute this z-value into the formula to find the corresponding x-value:

1.28 = (x - 82) / 8

Simplifying the equation:

1.28 * 8 = x - 82

10.24 = x - 82

x = 10.24 + 82

x = 92.24

Therefore, the score that separates the top 10% of the scores from the lowest 90% of the scores is approximately 92.24.