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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:
Where and
For this part we want to find a value a, such that we satisfy this condition:
(a)
(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:
But we know which value of z satisfy the previous equation so then we can do this:
And if we solve for a we got
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:
And if we solve for a we got
So the answer for this case would be (26.32; 57.68)