Respuesta :
Answer:
a) P ( X ≤ 49 ) = 0.9149
b) P ( X ≥ 48 ) = 0.2119
c) 0.9876
Explanation:
Solution:-
- Lets define a random variable X: the maximum speed of a moped.
- The random variable follows normal distribution with the following parameters:
X ~ Norm ( u , σ^2 )
Where,
u = Mean = 46.6 km/h
σ = Standard deviation 1.75 km/h
Hence,
X ~ Norm ( 46.6 , 1.75^2 ).
a) The probability that the maximum speed of mopeds is atmost 49 km/h?
- To evaluate the probability of P ( X ≤ 49 ). We will find the standard Z-score value using the following formula:
[tex]Z-score = \frac{x-u}{s} \\\\ Z-score = \frac{49-46.6}{1.75} \\\\Z-score = 1.37142[/tex]
- Now use the standard normal tables to determine the required probability of:
P ( Z < 1.37142 ) = 0.9149
Hence,
P ( X ≤ 49 ) = 0.9149
b) The probability that the maximum speed of mopeds is at-least 48 km/h?
- To evaluate the probability of P ( X ≥ 48 ). We will find the standard Z-score value using the following formula:
[tex]Z-score = \frac{x-u}{s} \\\\ Z-score = \frac{48-46.6}{1.75} \\\\Z-score = 0.8[/tex]
- Now use the standard normal tables to determine the required probability of:
P ( Z ≥ 0.8 ) = 0.2119
Hence,
P ( X ≥ 48 ) = 0.2119
c) The probability that the maximum speed of mopeds differs from mean by at-most 2.5 standard deviation.
- The required probability is the standard error from the mean value "u" of 2.5 standard deviation.
- We don't need to evaluate the test statistics as we are already given the standard error about mean.
- Using, the standard normal Z-score: The required probability is:
P ( -2.5 < Z < 2.5 ) = 2*P ( Z < 2.5 ) - 1
= 2*0.9938 - 1
= 0.9876
