A bank has kept records of the checking balances of its customers and determined that the average daily balance of its customers is $300 with a standard deviation of $48. A random sample of 144 checking accounts is selected.

a. What is the probability that the sample mean will be more than $306.60?
b. What is the probability that the sample mean will be less than $308?
c. What is the probability that the sample mean will be between $302 and $308?
d. What is the probability that the sample mean will be at least $296?

a)

And using the complement rule we got:

b)

And using the standard normal table or excel we got:

c)

And using the follwowing difference we got:

d)

And using the complement rule we got:

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the daily balances of a population, and for this case we know the distribution for X is given by:

Where and

since the distribution for X is normal then the distribution for the sample mean is given by:

The standard error is given by

And we want this probability:

And using the complement rule we got:

Part b

And using the standard normal table or excel we got:

Part c

And using the follwowing difference we got:

Part d

And using the complement rule we got:

answer; ///i believe the answer is (d) procedures to control the quailty of your product;