The same calculus exam has a normal distribution with a mean of 76 and a standard deviation of 6. Find the percentage of students that scored between 65 and 90.

z = (65-76)/6

z = -1.83

z = (90-76)/6
z = 2.33

To find the percentage of students that scored between 65 and 90 on the calculus exam, we need to calculate the area under the normal distribution curve between these two scores.

1. First, let's standardize the scores using the standard score formula (z-score):
z = (x - μ) / σ

where:
z is the standard score,
x is the given score,
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.

For the lower score of 65:
z1 = (65 - 76) / 6
= -11 / 6
= -1.83

For the upper score of 90:
z2 = (90 - 76) / 6
= 14 / 6
= 2.33

2. Next, we need to find the cumulative probabilities corresponding to these z-scores. We can use a standard normal distribution table or a calculator.

The cumulative probability associated with z1 = -1.83 is P(Z < -1.83).
The cumulative probability associated with z2 = 2.33 is P(Z < 2.33).

For example, using a standard normal distribution table or calculator, we can find:
P(Z < -1.83) ≈ 0.0336
P(Z < 2.33) ≈ 0.9904

3. Since we want to find the percentage of students between these two scores, we need to subtract the cumulative probability associated with the lower score from the cumulative probability associated with the upper score:

P(-1.83 < Z < 2.33) = P(Z < 2.33) - P(Z < -1.83)
≈ 0.9904 - 0.0336
≈ 0.9568

4. Finally, we can convert this probability to a percentage by multiplying it by 100:

Percentage = 0.9568 * 100 ≈ 95.68%

Therefore, approximately 95.68% of students scored between 65 and 90 on the calculus exam.