Respuesta :
Answer:
a) 0.1038 = 10.38% probability that maximum speed is at most 49 km/h
b) 0.2451 = 24.51% probability that maximum speed is at least 48 km/h
c) 0.9876 = 98.76% probability that maximum speed differs from the mean value by at most 2.5 standard deviations
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Mean value 46.8 km/h and standard deviation 1.75 km/h.
This means that [tex]\mu = 46.8, \sigma = 1.75[/tex]
a. What is the probability that maximum speed is at most 49 km/h?
This is the pvalue of Z when X = 49. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{49 - 46.8}{1.75}[/tex]
[tex]Z = 1.26[/tex]
[tex]Z = 1.26[/tex] has a pvalue of 0.8962
1 - 0.8962 = 0.1038
0.1038 = 10.38% probability that maximum speed is at most 49 km/h.
b. What is the probability that maximum speed is at least 48 km/h?
This is 1 subtracted by the pvalue of Z when X = 48. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{48 - 46.8}{1.75}[/tex]
[tex]Z = 0.69[/tex]
[tex]Z = 0.69[/tex] has a pvalue of 0.7549
1 - 0.7549 = 0.2451
0.2451 = 24.51% probability that maximum speed is at least 48 km/h.
c. What is the probability that maximum speed differs from the mean value by at most 2.5 standard deviations?
Zscores between Z = -2.5 and Z = 2.5, which is the pvalue of Z = 2.5 subtracted by the pvalue of Z = -2.5
Z = 2.5 has a pvalue of 0.9938
Z = -2.5 has a pvalue of 0.0062
0.9938 - 0.0062 = 0.9876
0.9876 = 98.76% probability that maximum speed differs from the mean value by at most 2.5 standard deviations
