Question

1.I recently asked 100 middle school students to complete a statistics test. The mean score on the test was 30 points with a standard deviation of 5 points. The scores followed a normal distributions. Using this information, calculate the following:

a. What is the probability a student earned a score of 45 points or less? P (score < 45 points) =

b. What is the probability a student earned a score higher than 30 points? P(score > 30) =

c. What is the probability a student earned a score between 25 and 45 points? P (25 points < score < 45 points) =

d. I want to know the cutoff value for the upper 10%. What score separates the lower 90% of scores from the upper 10%? P (score < _______) = 90% or 0.90 cumulative area to the left

e. I want to know the cutoff values for the lowest 25%. P (score < ________) = 25% or 0.25 cumulative area to the left

f. I would also like to know the cut off values for the highest 25%. P (score > _______) = 25% or 0.25 cumulative area to the right

Answer 1

Empirical rule:

The empirical rule states that for a normal distribution,  

• 68% of data falls within the first standard deviation from the mean.
• 95% fall within two standard deviations.
• 99.7% fall within three standard deviations.

For the given problem,

X: The scores

The scores : X follow  a normal distributions with mean = 30 and standard deviation : =5

a.

Probability a student earned a score of 45 points or less

Using empirical rule ; P(score < 45) = P(X<45) = P(X<) =1 - P(X>)=1-0.0015 = 0.9985

(0.15% = 0.15/100=0.0015)

P (score < 45 points) = 0.9985

Using z-score

Z-score for 45 = (45-30)/5 = 3 ; From normal tables, P(Z<3) = 0.9987;P(Z<3)=P(X<45)=0.9987

P(Score < 45 points) = 0.9987

b. What is the probability a student earned a score higher than 30 points? P(score > 30)

Using empirical rule ; P(score > 30) = P(X>30) = P(X<) =0.5

P(X>30) = 1 - P(X<30)

Using z-score

Z-score for 30 = (30-30)/5 = 0 ; From normal tables, P(Z<0) = 0.5;P(Z<3)=P(X<30)=0.5000

P(X>30) = 1 - P(X<30)=0-0.5=0.5

P(Score >30) =0.5

c. What is the probability a student earned a score between 25 and 45 points? P (25 points < score < 45 points)

P (25 points < score < 45 points) = P()

From the picture:

P() =(34/100)+(34/100)+(13.5/100)+(2.35/100)=0.8385

P (25 points < score < 45 points) = 0.8385

using z-score

P (25 points < score < 45 points) =P(25<X<45) = P(X<45)-P(X<25)

Z-score for 45 = (45-30)/5 =3 ; Z-score for 25 = (25-30)5 = -1

From standard normal tables, P(Z<3) =0.9987 P(Z<-1) = 0.1587

P(X<45)=P(Z<3) =0.9987 ; P(X<25)=P(Z<-1) = 0.1587

P(25<X<45) = P(X<45)-P(X<25) = 0.9987-0.1587=0.84

P (25 points < score < 45 points) =  P (25 points < score < 45 points) =0.84

d. I want to know the cutoff value for the upper 10%. What score separates the lower 90% of scores from the upper 10%? P (score < _______) = 90% or 0.90 cumulative area to the left

Let that score be X1

i.e

P(Score < X1) = 0.90 ; P(X<X1)=0.90

Z1 be z-score for X1 ;

Z1=(X1-30)/5 ; X1=30+5Z1

P(Z<Z1)=P(X<X1)=0.90

From standard normal tables,

P(Z<1.28) = 0.8997 0.90

Z1 = 1.28

X1=30+5Z1 = 30 + 5 x 1.28=30+6.4 = 36.4

P (score < 36.4) = 90%

e. I want to know the cutoff values for the lowest 25%. P (score < ________) = 25% or 0.25 cumulative area to the left

Let that score be X2

i.e

P(Score < X2) = 0.25 ; P(X<X2)=0.25

Z2 be z-score for X2 ;

Z2=(X2-30)/5 ; X2=30+5Z2

P(Z<Z2)=P(X<X2)=0.25

From standard normal tables,

P(Z<-0.67) = 0.2514

P(Z<-0.68) = 0.2483

Therefore, (-0.67+(-0.68))/2=-0.675; P(Z<=-0.675) 0.25

Z2 = -0.675

X2=30+5Z2 = 30 + 5 x -0.675 =30-3.375=26.625

P (score < 26.625) = 25%

f. I would also like to know the cut off values for the highest 25%. P (score > _______) = 25% or 0.25 cumulative area to the right

Let that score be X3

i.e

P(Score < X3) = 0.25 ; P(X<X3)=0.25;

P(X<X3)=1-P(X<X3) =0.25; P(X<X3) =1-0.25=0.75

Z3 be z-score for X3 ;

Z3=(X3-30)/5 ; X3=30+5Z3

P(Z<Z3)=P(X<X3)=0.75

From standard normal tables,

P(Z<0.67) = 0.7514

P(Z<0.68) = 0.7483

Therefore, (0.67+0.68)/2=0.675; P(Z<=0.675) 0.75

Z3 = 0.675

X3=30+5Z3 = 30 + 5 x 0.675 =30+3.375=33.375

P (score > 33.375) = 25%