Respuesta :
Answer:
(a) [tex]X\sim N(\mu=26,\ \sigma^{2}=13^{2})[/tex].
(b) The probability that a randomly selected LA worker has a commute that is longer than 34 minutes is 0.2676.
(c) The 70th percentile for the commute time of LA workers is 33 minutes.
Step-by-step explanation:
The random variable X is defined as the commute time for LA workers.
The mean commute time is, μ = 26 minutes and the standard deviation of the commute times is, σ = 13 minutes.
(a)
It is provided that the LA commute time fr workers is normally distributed.
Then the distribution of the random variable X can be defined as follows:
[tex]X\sim N(\mu=26,\ \sigma^{2}=13^{2})[/tex].
(b)
Compute the value of P (X > 34) as follows:
[tex]P(X>34)=P(\frac{X-\mu}{\sigma}>\frac{34-26}{13})[/tex]
[tex]=P(Z>0.62)\\=1-P(Z<0.62)\\=1-0.73237\\=0.26763\\\approx 0.2676[/tex]
*Use a z-table.
Thus, the probability that a randomly selected LA worker has a commute that is longer than 34 minutes is 0.2676.
(c)
The pth percentile is a data value such that at least p% of the data set is less than or equal to this data value and at least (100 - p)% of the data set are more than or equal to this data value.
The 70th percentile for the commute time of LA workers can be written as follows:
P (X < x) = 0.70
⇒ P (Z < z) = 0.70
The value of z for this probability is:
z = 0.53
*Use a z-table.
Compute the value of x as follows:
[tex]z=\frac{x-\mu}{\sigma}\\\\0.53=\frac{x-26}{13}\\\\x=26+(0.53\times 13)\\\\x=32.89\\\\x\approx 33[/tex]
Thus, the 70th percentile for the commute time of LA workers is 33 minutes.
