The mile run is a popular track event. In the Olympics, the races are in meters and the closest race to a mile is the 1500-meter race. If a meter is equal to a yard plus one-twelfth of a yard plus one-hundredth of a yard and there are 1760 yards in a mile, then how many yards further is the mile than the 1500-meter race?

Step-by-step explanation:

a)

Given that ; X ~ N ( µ = 65 , σ = 4 )P ( 64 < X < 66 )

From application of normal distribution ;

Z = ( X - µ ) / σ, Z = ( 64 - 65 ) / 4,  Z = -0.25Z = ( 66 - 65 ) / 4,  Z = 0.25

Hence, P ( -0.25 < Z < 0.25 ) = P ( 64 < X < 66 ) = P ( Z < 0.25 ) - P ( Z < -0.25 ) P ( 64 < X < 66 ) = 0.5987 - 0.4013

P ( 64 < X < 66 ) = 0.1974

b) X ~ N ( µ = 65 , σ = 4 )

P ( 64 < X < 66 )

From normal distribution application ;

Z = ( X - µ ) / ( σ / √(n)), plugging in the values, Z = ( 64 - 65 ) / ( 4 / √(12)) = Z = -0.866Z = ( 66 - 65 ) / ( 4 / √(12)) = Z = 0.866

P ( -0.87 < Z < 0.87 )

P ( 64 < X < 66 ) = P ( Z < 0.87 ) - P ( Z < -0.87 )P ( 64 < X < 66 ) = 0.8068 - 0.1932P ( 64 < X < 66 ) = 0.6135

c) From the values gotten for (a) and (b), it is indicative that the probability in part (b) is much higher because the standard deviation is smaller for the x distribution.