Question
he following figure is a probability density curve that represents the grade point averages (GPA) of the graduating seniors at a large university.
2.2 2.55 2.9 the area is 0.65
2.9 to 3.25 the area is 0.23
Find the proportion of seniors whose GPA is between 2.9 and 3.25
What is the probability that a randomly chosen senior will have a GPA greater than 3.25?.
2.2 2.55 2.9 the area is 0.65
2.9 to 3.25 the area is 0.23
Find the proportion of seniors whose GPA is between 2.9 and 3.25
What is the probability that a randomly chosen senior will have a GPA greater than 3.25?.
Answers
GPT-4o mini
To find the proportion of seniors whose GPA is between 2.9 and 3.25, we can directly use the information given in the question. The area under the probability density curve between 2.9 and 3.25 represents this proportion.
From the information provided:
- The area (proportion) between 2.9 and 3.25 is **0.23**.
So, the proportion of seniors whose GPA is between 2.9 and 3.25 is **0.23**.
Next, to find the probability that a randomly chosen senior will have a GPA greater than 3.25, we can use the following logic:
1. The total area under the probability density curve must equal 1 (since the total probability must be 1).
2. We already know the area (probability) for the interval \(2.9\) to \(3.25\) is \(0.23\) and the area from \(2.2\) to \(2.9\) is \(0.65\).
Let’s calculate the area from \(2.2\) to \(3.25\):
- The area from \(2.2\) to \(3.25\) is the sum of the areas from \(2.2\) to \(2.9\) and from \(2.9\) to \(3.25\):
\[
\text{Area from 2.2 to 3.25} = 0.65 + 0.23 = 0.88.
\]
3. Now, to find the area (probability) for GPA greater than \(3.25\):
\[
\text{Area greater than 3.25} = 1 - \text{Area from 2.2 to 3.25} = 1 - 0.88 = 0.12.
\]
Thus, the probability that a randomly chosen senior will have a GPA greater than \(3.25\) is **0.12**.
From the information provided:
- The area (proportion) between 2.9 and 3.25 is **0.23**.
So, the proportion of seniors whose GPA is between 2.9 and 3.25 is **0.23**.
Next, to find the probability that a randomly chosen senior will have a GPA greater than 3.25, we can use the following logic:
1. The total area under the probability density curve must equal 1 (since the total probability must be 1).
2. We already know the area (probability) for the interval \(2.9\) to \(3.25\) is \(0.23\) and the area from \(2.2\) to \(2.9\) is \(0.65\).
Let’s calculate the area from \(2.2\) to \(3.25\):
- The area from \(2.2\) to \(3.25\) is the sum of the areas from \(2.2\) to \(2.9\) and from \(2.9\) to \(3.25\):
\[
\text{Area from 2.2 to 3.25} = 0.65 + 0.23 = 0.88.
\]
3. Now, to find the area (probability) for GPA greater than \(3.25\):
\[
\text{Area greater than 3.25} = 1 - \text{Area from 2.2 to 3.25} = 1 - 0.88 = 0.12.
\]
Thus, the probability that a randomly chosen senior will have a GPA greater than \(3.25\) is **0.12**.