Answer:
The probability is [tex]P(X \le 70 ) = 0.7054 [/tex]
Step-by-step explanation:
From the question we are told that
The mean is [tex]\mu = 72.6 \ miles /hour [/tex]
The standard deviation is [tex]\sigma = 4.78 \ miles /hour[/tex]
The speed limit is [tex]x = 70 \ miles /hour[/tex]
Generally the probability that 5 cars pass and none are speeding is equivalent to the probability that the speed of the cars is less than x which is mathematically represented as
[tex]P(X \le x ) = 1 - P(X > x )[/tex]
Now
[tex]P(X > x) = P(\frac{X - \mu}{\sigma } >\frac{x - \mu}{\sigma } )[/tex]
Generally
[tex] \frac{X - \mu}{\sigma } = Z (The \ standardized \ value \ of \ X)[/tex]
So
[tex]P(X > 70) = P(Z >\frac{70 - 72.6}{72.6 } )[/tex]
[tex]P(X > 70) = P(Z > -0.54 )[/tex]
From the z-table
[tex] P(Z > -0.54 ) = 0.2946[/tex]
=> [tex]P(X > 70) = 0.2946 [/tex]
So
[tex]P(X \le 70 ) = 1 - P(X > 70 ) [/tex]
=> [tex]P(X \le 70 ) = 1 - 0.2946 [/tex]
=> [tex]P(X \le 70 ) = 0.7054 [/tex]
