SOLUTION: Furnace repair bills are normally distributed with a mean of 271 dollars and a standard deviation of 20 dollars. If 64 of these repair bills are randomly selected, find the probabi

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Question 1142020: Furnace repair bills are normally distributed with a mean of 271 dollars and a standard deviation of 20 dollars. If 64 of these repair bills are randomly selected, find the probability that they have a mean cost between 271 dollars and 273 dollars.
A) 0.2881
B) 0.5517
C) 0.2119
D) 0.7881
Answer by VFBundy(438) About Me  (Show Source):
You can put this solution on YOUR website!
We want to find the standard deviation (SD) of the sample. We do this by taking the SD of the population and dividing it by the square root of the number of items in the sample:

20%2Fsqrt%2864%29 = 20%2F8 = 2.5

Next, we figure out how many SDs 273 is above/below the mean. 273 is 2 above the mean of 271. So, we take +2 and divide it by the SD of the sample (2.5). This result is +0.80. So, we look up +0.80 on a z-table and find the result is 0.7881. This means there is a 0.7881 probability that the mean cost is below $273.

Now, we figure out how many SDs 271 is above/below the mean. 271 is identical to the mean of 271...meaning 271 is 0 above the mean. We divide 0 by the SD of the sample (2.5), and obviously come up with a result of 0. When we look up 0 on a z-table, we find the result is 0.5000. This means there is a 0.5000 probability that the mean cost is below $271.

To find the probability that the mean cost is between $271 and $273, simply subtract the probability that the mean cost is below $271 (0.5000) from the probability that the mean cost is below $273 (0.7881), to get a result of 0.2881.

So...the answer is (A).