According to a government study among adults in the 25- to 34-year age group, th

Question

According to a government study among adults in the 25- to 34-year age group, the mean amount spent per year on reading and entertainment is $2,035. Assume that the distribution of the amounts spent follows the normal distribution with a standard deviation of $601. (Round z-score computation to 2 decimal places and your final answer to 2 decimal places.)
  
(a)   What percent of the adults spend more than $2,475 per year on reading and entertainment?
  
Percent ?
  
(b)   What percent spend between $2,475 and $3,150 per year on reading and entertainment?
  
Percent ?
  
(c)   What percent spend less than $1,025 per year on reading and entertainment?
  
Percent ?

Explanation / Answer

The method to solving all parts of this problem is the normalcdf function on the graphing calculator. I'll be explaining the process in terms of a TI-84, but the method should be similar, if not identical for many of the graphing calculators out there.

To get to the normalcdf function on the calculator, we go to 2nd>VARS>2 (normalcdf). The screen will then prompt you to put in a lower bound, an upper bound, an average (displayed as the Greek letter mu), and a standard deviation (displayed as the Greek letter sigma). The last two - average and standard deviation - will be identical for all three parts of this problem because the curve on which all three parts are based is the same. The mean and the standard deviation will be 2035 and 601, rspectively. The upper and lower bounds differ depending on the question

Part A: Since the question asks for more than 2475, this value will be our lower bound and we will use 1E99 (eseentially infinity) as our upper bound. Entering these values and pressing Enter a few times will give:

normalcdf(2475, 1E99, 2035, 601) = .2320496815 which rounds to .2320, or 23.20%

Part B: Since the question asks for between 2475 and 3150, these will be our upper and lower bounds respectively. This yields:

normalcdf(2475, 3150, 2035, 601) = .200268853 which rounds to .2003 or 20.03%

Part C: Since the question asks for less than 1025, we will be using -1E99 (negative infinitiy) as our lower bound and 1025 as the upper bound. This will give us:

normalcdf(-1E99, 1025, 2035, 601) = .0464268577 which rounds to .0464 or 4.64%

Hope this helps!

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