can someone please explain the steps:
suppose that the grade point average,denoted by X,of a university student is Normally distributed with the mean 2.786 and standard deviation 0.375
A). a student is on the deans list if her GPA exceeds 3.746. What percent of the students are on the dean's list? P(X>3.746 GPA)
B). a student is on probation if his GPA falls below 1.946. what percent of the students are on probation?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores. Multiply by 100 to convert to percent.
To find the percentage of students on the Dean's list and on probation, we need to use the properties of the normal distribution. Here are the steps to find the answers:
A) To find the percentage of students on the Dean's list (P(X > 3.746 GPA)), we can use the cumulative probability function of the normal distribution.
Step 1: Standardize the value of 3.746 GPA using the formula:
Z = (X - μ) / σ
Where X is the value we want to standardize, μ is the mean of the normal distribution, and σ is the standard deviation.
In this case, X = 3.746, μ = 2.786, and σ = 0.375. So, we can calculate:
Z = (3.746 - 2.786) / 0.375
Step 2: Look up the standardized value (Z) in the standard normal distribution table or use a calculator or statistical software to find the cumulative probability.
Step 3: Subtract the cumulative probability from 1 to find the probability of being on the Dean's list:
P(X > 3.746 GPA) = 1 - P(Z < Z-value)
B) To find the percentage of students on probation (P(X < 1.946 GPA)), we follow a similar process as above:
Step 1: Standardize the value of 1.946 GPA using the formula:
Z = (X - μ) / σ
Where X is the value we want to standardize, μ is the mean of the normal distribution, and σ is the standard deviation.
In this case, X = 1.946, μ = 2.786, and σ = 0.375. So, we can calculate:
Z = (1.946 - 2.786) / 0.375
Step 2: Look up the standardized value (Z) in the standard normal distribution table or use a calculator or statistical software to find the cumulative probability.
P(X < 1.946 GPA) = P(Z < Z-value)
With these steps, you can calculate the percentage of students on the Dean's list and on probation.