The mean grade in this class last semester was 78.3, and the variance was 34.6 . The distribution of grades was unimodal and symmetrical.. determine how many students are likely to fail the class (<60), and how many students are likely to get a “A” (>90), if 60 students are enrolled. Show all your work, and provide a sentence explaining each step of your calculations.
"Unimodal and symmetrical" = normal
Z = (score-mean)/SD
variance = SD^2
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores. Multiply by 60.
So would I do this? 60-78.3/5.882 and 90-78.3/5.882
and multiply my answer by 60?
meaning I am multiplying the z-score by 60 to find the percentage/probabitlity?
To determine the number of students likely to fail the class (<60) and the number of students likely to get an "A" (>90), we need to use the information about the mean, variance, and distribution of grades in the class.
Step 1: Find the standard deviation (σ) of the class.
The standard deviation is the square root of the variance.
σ = √34.6 ≈ 5.88
Step 2: Calculate the z-scores for the grades of 60 and 90.
The z-score tells us how many standard deviations a particular grade is away from the mean. We can use it to determine the likelihood of getting a certain grade.
z = (x - μ) / σ
For x = 60:
z1 = (60 - 78.3) / 5.88 ≈ -3.11
For x = 90:
z2 = (90 - 78.3) / 5.88 ≈ 1.98
Step 3: Use a standard normal distribution table or a calculator to find the probabilities associated with the z-scores.
The standard normal distribution table provides the probability of getting a score less than or equal to a certain z-score.
For z1 = -3.11:
P(Z ≤ -3.11) ≈ 0.0008 or 0.08% (This is the probability of getting a score less than 60.)
For z2 = 1.98:
P(Z ≤ 1.98) ≈ 0.9762 or 97.62% (This is the probability of getting a score less than or equal to 90.)
Step 4: Calculate the number of students likely to fail and get an "A".
To find the number of students likely to fail (<60), we multiply the probability by the total number of students:
Number of students likely to fail = 0.08% * 60 ≈ 0.048 or about 3 students.
To find the number of students likely to get an "A" (>90), we multiply the probability by the total number of students:
Number of students likely to get an "A" = 97.62% * 60 ≈ 58.57 or about 59 students.
Therefore, it is likely that around 3 students will fail the class (<60), and about 59 students will get an "A" (>90) out of the 60 students enrolled.