SOLUTION: A car insurance plan has the expected mean annual payment to a client μ = $950 and the expected standard deviation of the annual payments to clients σ = $4,220. The insurance com

Question 1199105: A car insurance plan has the expected mean annual payment to a client μ = $950 and the expected standard deviation of the annual payments to clients σ = $4,220. The insurance company charges $960 per year for that insurance plan. There are 500,000 clients, who bought this plan. What is the probability that this plan will be profitable in the upcoming year?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i'm not exactly sure how to look at this.
i'm thinking as follows:

the population is all the clients of the insurance policy.
the mean payment to the clients is 950 per year.
the standard deviation to the clients is 4220 per year.
that would be your population mean and standard deviation.
in order for the plan to be profitable, the cost per client needs to be less than 960 per year.
you would need to find the area under the normal distribution curve (assuming the distribution is normal) that is to the left of 960.
use the z-score to find this.
z-score formula is z = (x - m) / s
z is the z-score
x is the revenue per year.
m is the mean of the cost per year.
s is the standard deviation of the cost per year.
you get z = (960 - 950) / 4220 = .0023696682
area to the left of that z-score is equal to .5009453664
that says that 50.09% of the cost per person is less than 960 per year.
what that says is that the probability that the plan will be profitable is 50.09%.
see if this works for you.
let me know what you think.

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