Question

The amount of time that a​ drive-through bank teller spends on a customer is a random variable with a mean mu = 4.9 minutes and a standard deviation sigma = 2.4 minutes. If a random sample of 36 customers is​ observed, find the probability that their mean time at the​ teller's window is

​(a) at most 4.3 ​minutes;

​(b) more than 5.3 ​minutes;

​(c) at least 4.9 minutes but less than 5.7 minutes.

(a) The probability that the mean time is at most 4.3 minutes is ___. ​(Round to four decimal places as​ needed.)

​(b) The probability that the mean time is more than 5.3 minutes is ___. ​(Round to four decimal places as​ needed.)

​(c) The probability that the mean time is between 4.9 minutes and 5.7 minutes is ___ . ​(Round to four decimal places as​ needed.)

Answer 1

Solution :

Given that ,

= 4.9

= / n = 2.4 / 36 = 0.4

a) P( 4.3 ) = P(( - ) / (4.3 - 4.9) / 0.4)

= P(z -1.5)

Using z table

= 0.0668

b) P( > 5.3) = 1 - P( < 5.3)

= 1 - P[( - ) / < (5.3 - 4.9) / 0.4 ]

= 1 - P(z < 1.0)   

= 1 - 0.8413

= 0.1587

c) P(4.9 < < 5.7 )  

= P[(4.9 - 4.9) / 0.4 < ( - ) / < (5.7 - 4.9) / 0.4 )]

= P(0 < Z < 2.0)

= P(Z < 2.0) - P(Z < 0)

Using z table,  

= 0.9772 - 0.5

= 0.4772