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?

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:

[tex]X \sim N(300,48)[/tex]

Where [tex]\mu=300[/tex] and [tex]\sigma=48[/tex]

since the distribution for X is normal then the distribution for the sample mean [tex]\bar X[/tex] is given by:

[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

The standard error is given by [tex]\sigma_{bar X}=\frac{48}{\sqrt{144}}= 4[/tex]

And we want this probability:

[tex] P(\bar X> 306.6) = P(Z> \frac{306.6-300}{4}) = P(Z>1.65)[/tex]

And using the complement rule we got:

[tex] P(Z>1.65)= 1-P(Z<1.65) =1-0.951= 0.049[/tex]

Part b

[tex]P(\bar X<308) = P(z<\frac{308-300}{4}) = P(Z<2)[/tex]

And using the standard normal table or excel we got:

[tex]P(Z<2)= 0.97725[/tex]

Part c

[tex] P(302< \bar X <308) = P(\frac{302-300}{4}<Z<\frac{308-300}{4}) = P(0.5<Z<2)[/tex]

And using the follwowing difference we got:

[tex] P(0.5<Z<2) = P(Z<2) -P(Z<0.5) = 0.97725-0.691= 0.286[/tex]

Part d

[tex]P(\bar X>296) = P(z>\frac{296-300}{4}) = P(Z>-1)[/tex]

And using the complement rule we got:

[tex] P(Z>-1)= 1-P(Z<-1) = 1-0.159=0.841[/tex]