Question

# Scores of students in a large Statistics class test are normally distributed with a mean of...

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

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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