Question

It has been reported that last year 25% of drivers have fallen asleep at the wheel....

It has been reported that last year 25% of drivers have fallen asleep at the wheel. Source: National Sleep Foundation If 100 drivers are selected at random, find the following probabilities: [4a] Find the probability that at most 27 drivers will say that they have fallen asleep while driving. [4b] Use [4a] to find the probability that exactly 27 drivers will say that they have fallen asleep while driving. [4c] Find the probability that less than 24 drivers will say that they have fallen asleep while driving. [4d] Use [4a, b, c] to find the probability that between 24 and 27 drivers will say that they have fallen asleep while driving.

 here mean of distribution=μ=np= 25.00 and standard deviation σ=sqrt(np(1-p))= 4.33 for normal distribution z score =(X-μ)/σx
 therefore from normal approximation of binomial distribution and continuity correction:

a)

 probability =P(X<27.5)=(Z<(27.5-25)/4.33)=P(Z<0.58)=0.7190

b)

 probability =P(26.5

c)

 probability =P(X<23.5)=(Z<(23.5-25)/4.33)=P(Z<-0.35)=0.3632

d)

 probability =P(23.5