Respuesta :
Answer:
(26.32; 57.68)
See explanation below.
Step-by-step 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:
[tex]X \sim N(42,8)[/tex]
Where [tex]\mu=42[/tex] and [tex]\sigma=8[/tex]
For this part we want to find a value a, such that we satisfy this condition:
[tex]P(X>a)=0.975[/tex] (a)
[tex]P(X<a)=0.025[/tex] (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:
[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.025[/tex]
[tex]P(z<\frac{a-\mu}{\sigma})=0.025[/tex]
But we know which value of z satisfy the previous equation so then we can do this:
[tex]z=-1.96<\frac{a-42}{8}[/tex]
And if we solve for a we got
[tex]a=42 -1.96*8=26.32[/tex]
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:
[tex]z=1.96<\frac{a-42}{8}[/tex]
And if we solve for a we got
[tex]a=42 +1.96*8=57.68[/tex]
So the answer for this case would be (26.32; 57.68)
