Scores of students in a large Statistics class test are normally distributed with a mean of 75 points and a standard deviation of 5 points.Find the probability that a student scores more than 80 pointsIf 100 students are picked at random, how many do you expect to score below 70 points?If top 10% of students obtain an A grade, how much should a student obtain to get an A grade.Suppose four students are picked at random, find the probability that the average score of those four students is more than 80 points.
Solution :
Given that ,
mean =
= 75
standard deviation =
= 5
P(x > 80)
= 1 - P( x < 80)
=1 - P[(x -
) /
< (80 - 75) / 5]
=1 - P(z < 1)
Using z table,
= 1 - 0.8413
= 0.1587
Probability = 0.1587
n = 100
= 75
=
/
n = 5 /
100 = 0.5
P( < 70)
= P((
-
) /
< (70 - 75) / 0.5)
= P(z < -10)
Using z table
= 0.00
Probability = 0.00
The z-distribution of the 10% is,
P(Z > z) = 10%
= 1 - P(Z < z ) = 0.10
= P(Z < z ) = 1 - 0.10
= P(Z < z ) = 0.90
= P(Z < 1.282 ) = 0.90
z = 1.282
Using z-score formula,
x = z *
+
x = 1.282 * 5 + 75
x = 81.41
Answer :- 81 Students.
n = 4
= 75
=
/
n = 5 /
4 = 2.5
P(
> 80) = 1 - P(
< 80)
= 1 - P[(
-
) /
< (80 - 75) / 2.5]
= 1 - P(z < 2)
Using z table,
= 1 - 0.9772
= 0.0228
Probability = 0.0228
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