Answer:
A.) Probability of scoring 80 and above = 0.3881, Number of students = 310 B.) Probability of scoring 75 and below = 0.3336, Number of students = 268 C.) Probability of scoring from 71 to 85 = 0.6826, Number of students = 546
Step-by-step explanation:
Given: Mean (μ) = 78 Standard deviation (σ) = 7 Total number of students = 800
A.) To find the probability of students scoring 80 and above: First, we need to find the Z-score for a score of 80:
Z = X-μ/σ = 80-78/7 = 2/7 = 0.2857
Now, we find the probability using a Z-table: ( > 0.2857) = 1 − ( < 0.2857)
P ( Z > 0.2857) = 1− P(Z<0.2857) (>0.2857) ≈ 1 −0.6119 ≈ 0.3881
To find the number of students who scored 80 and above:
= ∗
= 800 ∗ 0.3881 ≈ 310.48 ≈ 310
B.) To find the probability of students scoring 75 and below: Using the Z-score formula:
Z = X- μ/σ = 75-78/7 = -0.4286
Now, we find the probability using a Z-table:
P ( Z < −0.4286) ≈ 0.3336
To find the number of students who scored 75 and below:
= ∗
= 800 ∗ 0.3336 ≈ 267.68 ≈ 268
C.) To find the probability of students scoring from 71 to 85: We need to find the Z-scores for 71 and 85:
For 71: Z = 71-78/7 = -1
For 85: Z = 85-78/7 = 1
Now, find the probability of Z being between -1 and 1 using the Z-table:
P ( −1 < Z <1 ) ≈ P ( Z < 1 ) − P ( Z < −1 ) = 0.8413 −0.1587 =0.6826
To find the number of students who scored from 71 to 85:
= ∗
= 800 ∗ 0.6826 ≈ 546.08 ≈ 546
