Finding the probability that the temperature is between x and y

Hello I have a problem that I can't seem to come to the proper conclusion with. It is as follows:

What is the probability that a randomly selected day in August will have a temperature greater than 85 but less than 100. Mean is 80 with a std dev of 8 assuming temperature is distributed normally.

I missed this on a test and I'd like to know where I went wrong so if someone could work it out step by step I would appreciate it very much. Thanks

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Are we assuming temperatures are normally distributed? – Gerry Myerson Oct 31 '12 at 1:57
Yes my bad. Assume that the temperatures are normally distributed. – Chris Maness Oct 31 '12 at 1:58

2 Answers

If your $\mu$ is 80, and $\sigma$ is 8, you're looking for $$P(85\le X \le 100)$$ Which can be found like this: $P(X\le 100)-P(X\le 85)$, letting $X=\frac{x-\mu}{\sigma}$ when you sub in your given info you can use the standard normal table and get your $z$ statistic (as I have been told they're called). So now you're looking to get $P(z \le 2.5)-P(z \le .625)$, and I'm assuming you can do the rest.

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Yeah, I do the following: Find P(Z < 2.5) = 0.99379 Find P(Z < 0.625) = 0.734014 So then I subtract and get 0.2598 This is exactly what I put on the test. However, it was marked wrong. – Chris Maness Oct 31 '12 at 2:33
Then it's time to ask your instructor about the problem. – Gerry Myerson Oct 31 '12 at 4:53
@Chris you should let us know so we know whether we had the right answer or not ;) – TheHopefulActuary Nov 5 '12 at 18:42

If you believe the distribution is normal (which it is not) you are asking for the probability that you are between 0.625 standard deviations and 2.5 standard deviations above the mean. You should have a table of the cumulative error function that will give you the answer. (For the Wikipedia table you will have to interpolate between 0.62 and 0.63)

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