Question

Suppose that student scores on math skills test are normally distributed. The mean of the test is 35 and the standard deviation is 4. Using a z-table (normal curve table), what percentage of students have z-scores a) below 2.05 b) above -0.50 Using a z-table, what scores would be the top and bottom score to find the c) middle 15% of students d) middle 25% of students Using a z-table, what is the minimum raw score a student can have on the math skills test and be in the bottom e) 25% f) 5%

Answer 1

Part a)

Z = 2.05
P ( Z < 2.05 )  = 0.9798
Percentage = 97.98%

Part b)

P ( Z > -0.5 ) = 1 - P ( Z < -0.5 )
P ( Z > -0.5) = 1 - 0.3085
P ( Z > -0.5) = 0.6915

Percentage = 69.15%

Part c)

X ~ N ( µ = 35 , σ = 4 )
P ( a < X < b ) = 0.15
Dividing the area 0.15 in two parts we get 0.15/2 = 0.075
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.075
Area above the mean is b = 0.5 + 0.075
Looking for the probability 0.425 in standard normal table to calculate Z score = -0.1891
Looking for the probability 0.575 in standard normal table to calculate Z score = 0.1891
Z = ( X - µ ) / σ
-0.1891 = ( X - 35 ) / 4
a = 34.2436
0.1891 = ( X - 35 ) / 4
b = 35.7564
P ( 34.2436 < X < 35.7564 ) = 0.15

Part d)

X ~ N ( µ = 35 , σ = 4 )
P ( a < X < b ) = 0.25
Dividing the area 0.25 in two parts we get 0.25/2 = 0.125
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.125
Area above the mean is b = 0.5 + 0.125
Looking for the probability 0.375 in standard normal table to calculate Z score = -0.3186
Looking for the probability 0.625 in standard normal table to calculate Z score = 0.3186
Z = ( X - µ ) / σ
-0.3186 = ( X - 35 ) / 4
a = 33.7256
0.3186 = ( X - 35 ) / 4
b = 36.2744
P ( 33.7256 < X < 36.2744 ) = 0.25

Part e)

X ~ N ( µ = 35 , σ = 4 )
P ( X < x ) = 25% = 0.25
To find the value of x
Looking for the probability 0.25 in standard normal table to calculate Z score = -0.6745
Z = ( X - µ ) / σ
-0.6745 = ( X - 35 ) / 4
X = 32.302
P ( X < 32.302 ) = 0.25

Part f)

X ~ N ( µ = 35 , σ = 4 )
P ( X < x ) = 5% = 0.05
To find the value of x
Looking for the probability 0.05 in standard normal table to calculate Z score = -1.6449
Z = ( X - µ ) / σ
-1.6449 = ( X - 35 ) / 4
X = 28.4204
P ( X < 28.4204 ) = 0.05