Answer:
P = 95%
Step-by-step explanation:
The average is:
[tex]\mu = 3\ minutes[/tex]
The standard deviation is:
[tex]\sigma = 0.25\ minutes[/tex].
We want the probability that the red light lasts between 2.5 minutes and 3.5 minutes
This is:
[tex]P(2.5 <X <3.5)[/tex]
Now we must transform these values to those of a standard normal distribution to facilitate calculation by using the probability tables.
[tex]P(2.5-3 <X- \mu<3.5-3)\\\\P(\frac{2.5-3}{0.25} <\frac{X- \mu}{\sigma}<\frac{3.5-3}{0.25})\\\\P(-2<Z<2)[/tex]
This is:
[tex]P(-2 <Z <2) = P(Z <2) - P(Z <-2)[/tex] ---------- (By the symmetry of the standard normal distribution)
When you search for the normal standard table, you get the following value:
[tex]P(Z <2) = 0.9772\\\\P(Z <-2) = 0.0228\\\\P(-2 <Z <2) = 0.9772 - 0.0228\\\\P(-2 <Z <2) = 0.9544[/tex]
