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Determine the area under a normal distribution graph with a mean of 55 and a standard deviation of 7?

a) to the right of x = 58

b) to the left of x = 24

Please show graphs if you can!!

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1 Answer

We have to convert the x-values to z-values first, and then look up areas to the left of the z-values on a standard normal distribution table, and, once we have those areas, determine if they represent the area under the curve we need.

μ = 55, σ = 7 for our normal distribution; that is given.  We know the relationship between x and Z:

Z = ( x - μ ) / σ

To find the area to the right of x = 58, let's first convert x to Z: Z = ( 58 - 55 ) / 7.  Z = 3/7 or ~.43.  I'm rounding to the nearest hundredth since the Z table lists values to the nearest .01.

The area to the right of Z = 0.43 comes from the table, but there's something important to note: areas from the table are those to the LEFT of a Z value.  We need the area to the RIGHT.  How do we handle this?  We use the fact that the area under the entire curve equals 1, and if we subtract out an area to the left of a Z value from 1, the remaining area is that to the RIGHT of the Z value, which is the area we seek.

The area to the left of Z = 0.43 is 0.6664, so the area to the right is 1 - 0.6664, or 0.3336.

For part B, x = 24 converts to Z = ( 24 - 55 ) / 7.  Z = -31/7, or ~ -4.43.

Depending on how far your Z table goes, you may not be able to look this one up.  The table I'm using ends at -3.49 on the left, and that area is listed as 0.0002.  So the answer to part B is "less than or equal to 0.0002", since we can't look it up exactly.