Answer to Question #351020 in Statistics and Probability for lei

Question

the time between busses on stevens creek blvd is 12 minutes. Therefore the wait time of a passenger who arrives randomly at a bus stop is uniformly distributed between 0 and 12 minutes.

a. find the probability that a person randomly arriving at the bus stop to wait for the bus has a wait time of at most 5 minutes.

in the accompanying diagram, the shaded area represents approximately 95% of the scores on a standardized test. If these scores ranged from 78 to 92,

a. what is the mean?

b. what is the standard deviation?

Answer 1

a.

$$P(X\le 5)=\displaystyle\int_{0}^{5}\dfrac{1}{12}dx=\dfrac{5}{12}$$

b. The shaded area is symmetric with respect the line $x=\mu$

$$\mu=\dfrac{78+92}{2}=85$$

$$P(78<X<92)=P(Z<\dfrac{92-85}{\sigma})$$

$$-P(Z\le\dfrac{78-85}{\sigma})=P(Z<\dfrac{7}{\sigma})-P(Z\le -\dfrac{7}{\sigma})$$

$$=0.95$$

$$P(Z\le -\dfrac{7}{\sigma})=\dfrac{1-0.95}{2}$$

$$-\dfrac{7}{\sigma}=-1.96$$

$$\sigma=\dfrac{7}{1.96}$$

$$\sigma=3.57$$