Question

Pay your bills: In a large sample of customer accounts, a utility company determined that the average number of days between when a bill was sent out and when the payment was made is with a standard deviation of days. Assume the data to be approximately bell-shaped.

Answer 1

(26.32; 57.68)

See explanation below.

Explanation:

Assuming the following question:

Pay your bills:

In a large sample of customer accounts, a utility company determined that the average number of days between when a bill was sent out and when the payment was made is 42 with a standard deviation of 8 days. Assume the data to be approximately bell-shaped.

Between what two values will approximately 95% of the numbers of days be?

Solution to the problem

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".

Let X the random variable that represent the average number of days between when a bill was sent out and when the payment of a population, and for this case we know the distribution for X is given by:

`X \sim N(42,8)`

Where
`\mu=42` and
`\sigma=8`

For this part we want to find a value a, such that we satisfy this condition:

`P(X>a)=0.975` (a)

`P(X<a)=0.025` (b)

Both conditions are equivalent on this case. We can use the z score in order to find the value a.

As we can see on the figure attached the z value that satisfy the condition with 0.025 of the area on the left and 0.975 of the area on the right it's z=-1.96. On this case P(Z<-1.96)=0.025 and P(z>-1.96)=0.975

If we use condition (b) from previous we have this:

`P(X<a)=P((X-\mu)/(\sigma)<(a-\mu)/(\sigma))=0.025`

`P(z<(a-\mu)/(\sigma))=0.025`

But we know which value of z satisfy the previous equation so then we can do this:

`z=-1.96<(a-42)/(8)`

And if we solve for a we got

`a=42 -1.96*8=26.32`

So the value of height that separates the bottom 2.5% of data from the top 97.5% is 26.32.

For the other value since the distirbution is symmetric the other value that accumulates 0.975 of the area on the left and 0.025 on the right is z=1.96 and similarly:

`z=1.96<(a-42)/(8)`

And if we solve for a we got

`a=42 +1.96*8=57.68`

So the answer for this case would be (26.32; 57.68)