A district mathematics test for third-graders had a normal distribution with a mean of 74 and a standard deviation of 11. What percentage of students scored between 63 and 85?

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To find the percentage of students who scored between 63 and 85, we need to calculate the area under the normal distribution curve between those two scores.

First, let's standardize the scores by using the z-score formula:
z = (x - μ) / σ, where x is the score, μ is the mean, and σ is the standard deviation.

For 63: z = (63 - 74) / 11 = -1
For 85: z = (85 - 74) / 11 ≈ 1

Next, we need to find the area between the z-scores of -1 and 1. We can use a z-table or a calculator to look up the probability corresponding to these z-scores.

Using a z-table, the probability corresponding to a z-score of -1 is approximately 0.1587, and the probability corresponding to a z-score of 1 is approximately 0.8413.

The area between these two z-scores is:
0.8413 - 0.1587 = 0.6826

Therefore, approximately 68.26% of students scored between 63 and 85 on the test.

To find the percentage of students who scored between 63 and 85, we need to use the properties of a normal distribution.

Step 1: Standardize the scores
To standardize the scores, we need to convert them into z-scores using the formula:

z = (x - μ) / σ

Where:
x = the score
μ = the mean
σ = the standard deviation

For the lower end, which is 63:
z1 = (63 - 74) / 11
z1 = -1.00

For the upper end, which is 85:
z2 = (85 - 74) / 11
z2 = 1.00

Step 2: Find the area under the standard normal curve
Now that we have the z-scores, we can use a standard normal table or a calculator to find the area under the curve between these two z-scores. The area represents the percentage of students who scored between 63 and 85.

Using a standard normal table, we can find the area for each z-score:
The area to the left of z1 is 0.1587 and the area to the left of z2 is 0.8413.

To find the percentage between z1 and z2, subtract the area to the left of z1 from the area to the left of z2:
Area = 0.8413 - 0.1587
Area = 0.6826

Step 3: Convert the area to a percentage
To convert the area to a percentage, multiply it by 100:
Percentage = 0.6826 * 100
Percentage ≈ 68.26

Therefore, approximately 68.26% of students scored between 63 and 85 on the district mathematics test.