At a high school, GPA’s are normally distributed with a mean of 2.6 and a standard deviation of 0.5. What percentage of students at the college have a GPA between 2.1 and 3.1?
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It's a handy tool.
2.5
To find the percentage of students at the high school with a GPA between 2.1 and 3.1, you can use the standard normal distribution table or a calculator with a normal distribution function.
Step 1: Standardize the GPA values
To standardize the GPA values of 2.1 and 3.1, you need to convert them to z-scores using the formula:
z = (x - μ) / σ
Where:
x = GPA value
μ = mean of the GPA distribution (2.6)
σ = standard deviation of the GPA distribution (0.5)
For a GPA of 2.1:
z1 = (2.1 - 2.6) / 0.5
For a GPA of 3.1:
z2 = (3.1 - 2.6) / 0.5
Step 2: Determine the area under the standard normal curve
Next, you need to find the area under the standard normal curve between the two z-scores (z1 and z2) obtained in Step 1. This will give you the percentage of students with a GPA between 2.1 and 3.1.
You can look up the corresponding values from the standard normal distribution table or use a calculator with a normal distribution function to find the cumulative probability (area) between z1 and z2.
Step 3: Calculate the percentage
Once you have the area between z1 and z2, you can convert it to a percentage by multiplying it by 100.
The final result will give you the percentage of students at the high school with a GPA between 2.1 and 3.1.
To find the percentage of students at the college who have a GPA between 2.1 and 3.1, we can use the properties of the normal distribution.
First, we need to standardize the GPAs. Standardizing the values will convert them to z-scores, which represent the number of standard deviations a particular GPA is from the mean.
The formula to standardize a value using a mean (μ) and standard deviation (σ) is:
z = (x - μ) / σ
Where:
z is the standardized value (z-score),
x is the original value,
μ is the mean, and
σ is the standard deviation.
Now, let's calculate the z-scores for GPA values of 2.1 and 3.1.
For a GPA of 2.1:
z1 = (2.1 - 2.6) / 0.5
For a GPA of 3.1:
z2 = (3.1 - 2.6) / 0.5
Calculating these z-scores, we find:
z1 = -1.0
z2 = 1.0
Once we have the z-scores, we can look up the corresponding areas under the standard normal distribution curve using a standard normal distribution table or a calculator.
Since we want to find the percentage of students with GPAs between 2.1 and 3.1, we need to find the area under the curve between z1 and z2.
Using the standard normal distribution table or a calculator, the area to the left of z = -1.0 is 0.1587, and the area to the left of z = 1.0 is 0.8413.
To calculate the percentage of students with GPAs between 2.1 and 3.1, we subtract the area to the left of z1 from the area to the left of z2:
Percentage between 2.1 and 3.1 = 0.8413 - 0.1587 = 0.6826
Therefore, approximately 68.26% of students at the college have a GPA between 2.1 and 3.1.