The marks obtained by students in a Mathematics examination are found to be norm

Question

The marks obtained by students in a Mathematics examination are found to be normally distributed with a mean of 65 and a standard deviation of 15. The same students did an English test and the results were normally distributed with a mean of 53 and a standard deviation of 20.

a. Estimate the proportion of students scoring less than 50 in Mathematics.

b. What proportion of students performed better in English than in mathematics?

c. Students receiving a combined score of 150 or more are awarded a distinction. What proportion of students receive a distinction?

d. A special prize, based on the combined score in English and Mathematics, is awarded to the top 5% of the students. Determine the lowest combined score obtained by students in this group.

Explanation / Answer

a)proportion of students scoring less than 50 in Mathematics=P(X<50)=P(Z<(50-65)/15)=P(Z<-1)= 0.1587

b)

here let score in math is X and score in Y is english ;

therefore estimated mean X-Y =65-53=12

and std deviation of X-Y =(152+202)1/2 =25

therefore  proportion of students performed better in English than in mathematics =P(X-Y<0)

=P(Z<(0-12)/25)=P(Z<-0.48)=0.3156

c)

for expected mean of X+Y =65+53=118

std deviation of X+Y =(152+202)1/2 =25

proportion of students receive a distinction =P(X>150)=1-P(X<150)=1-P(Z<(150-118)/25)=1-P(Z<1.28)

=1-0.8997=0.1003

d)

for top 5% ; crtiical value of z =1.6449

hence coresponding combined score obtained =mean +z*Std deviation =118+1.6449*25=159.12

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