At one college, GPAs are normally distributed with a mean of 2.6 and a standard deviation of 0.6. What percentage of students at the college have a GPA between 2 and 3.2?

Well, statistically speaking, let me juggle some numbers for you. Since GPAs at the college are normally distributed, we can use the Z-score to determine the percentage.

Firstly, let's calculate the Z-score for a GPA of 2:

Z1 = (2 - 2.6) / 0.6 ≈ -1

Next, let's calculate the Z-score for a GPA of 3.2:

Z2 = (3.2 - 2.6) / 0.6 ≈ 1

The Z-scores tell us how many standard deviations a particular GPA is from the mean. Now, we need to find the area between these two Z-scores.

Using a Z-table (which is like a clown car for Z-scores), we can find that the area to the left of Z1 is approximately 0.1587, and the area to the left of Z2 is approximately 0.8413.

To find the percentage of students with GPAs between 2 and 3.2, we subtract the area to the left of Z1 from the area to the left of Z2:

0.8413 - 0.1587 = 0.6826

So, approximately 68.26% of students at the college have a GPA between 2 and 3.2. Keep in mind that these numbers are approximate, but hey, who doesn't love a bit of statistical clowning around?

To find the percentage of students with a GPA between 2 and 3.2, we need to calculate the area under the normal distribution curve between these two GPA values.

Step 1: Standardize the GPA values
To standardize a GPA value, we subtract the mean (2.6) from the GPA value and then divide by the standard deviation (0.6).

Standardized GPA for 2: (2 - 2.6) / 0.6 = -1
Standardized GPA for 3.2: (3.2 - 2.6) / 0.6 = 1

Step 2: Find the corresponding z-scores
Using a standard normal distribution table or a calculator, we can find the corresponding z-scores for -1 and 1. The z-score for -1 is approximately 0.1587, and the z-score for 1 is approximately 0.8413.

Step 3: Calculate the area between the z-scores
To find the percentage of students with a GPA between 2 and 3.2, we subtract the smaller z-score from the larger z-score: 0.8413 - 0.1587 = 0.6826.

Step 4: Convert into percentage
To convert the result into a percentage, we multiply by 100: 0.6826 * 100 = 68.26%.

Therefore, approximately 68.26% of students at the college have a GPA between 2 and 3.2.

To find the percentage of students at the college who have a GPA between 2 and 3.2, we can use the standard normal distribution.

Step 1: Standardize the values
To work with the standard normal distribution, we need to standardize the values using the formula:
Standardized value (z) = (x - mean) / standard deviation

For x = 2:
z1 = (2 - 2.6) / 0.6 = -1

For x = 3.2:
z2 = (3.2 - 2.6) / 0.6 = 1

Step 2: Find the area under the curve
Next, we can use a standard normal distribution table or a calculator to find the area under the curve between z1 and z2. This represents the percentage of students with a GPA between 2 and 3.2.

Using a standard normal distribution table, we find:
Area between z1 and z2 = Area(z < z2) - Area(z < z1)

For z = -1, the area is approximately 0.1587.
For z = 1, the area is approximately 0.8413.

So, the area between z1 and z2 is:
Area between z1 and z2 = 0.8413 - 0.1587 = 0.6826

Step 3: Convert the area to a percentage
Finally, we multiply the area by 100 to find the percentage of students.

Percentage of students with GPA between 2 and 3.2 = 0.6826 * 100 = 68.26%

Therefore, approximately 68.26% of students at the college have a GPA between 2 and 3.2.

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