Applied Statistics in Business and Economics/Finding z for a Given Area ?

2 Responses to “Applied Statistics in Business and Economics/Finding z for a Given Area ?”

1. cidyah says:

   October 21, 2010 at 3:40 pm

   From the Normal table, the z-value that corresponds to the upper 5 % is 1.64
   z=(x-mean)/sd
   1.64=(x-600)/100
   x=100(1.64)+600 =764

   b) The area above z is 0.75. The z value such that the area above z is 0.75 is -0.67
   -0.67 = (x-600)/100
   x=600-67=533

   c) The area from 0 to z is 0.40 (and the area below is 0.40)
   z=-1.28 and 1.28
   -1.28=(x-600)/100
   x=600-128=472
   and x=600+128=728
   The middle 80 % lies between 472 and 728.
2. Merlyn says:

   October 21, 2010 at 4:24 pm

   For any normal random variable X with mean μ and standard deviation σ, X ~ Normal(μ, σ), { note that in most textbooks and literature the notation is with the variance, i.e., X ~ Normal(μ, σ²). Most software denotes the normal with just the standard deviation.}

   You can translate into standard normal units by:
   Z = (X – μ) / σ
   form the Standard normal to X by:
   X = μ + Z σ

   Where Z ~ Normal(μ = 0, σ = 1). You can then use the standard normal cdf tables to get probabilities.

   If you are looking at the mean of a sample, then remember that for any sample with a large enough sample size the mean will be normally distributed. This is called the Central limit theorem.

   If a sample of size is is drawn from a population with mean μ and standard deviation σ then the sample average xBar is normally distributed

   with mean μ and standard deviation σ / √(n)

   An applet for finding the values
   http://www-stat.stanford.edu/~naras/jsm/FindProbability.html

   calculator
   http://stattrek.com/Tables/normal.aspx

   how to read the tables
   http://rlbroderson.tripod.com/statistics/norm_prob_dist_ed9.html

   in this case we need to translate from the standard normal to the variable given.

   for the upper 5% the z score is 1.645
   X = 600 + 1.645 * 100 = 764.50

   for the upper 75% we have a z score of -0.68
   X = 600 – 0.68 * 100 = 532.00

   for the middle 80% we have a zscore of -1.28 and 1.28
   X1 = 600 – 1.28 * 100 = 472.00
   X2 = 600 + 1.28 * 100 = 728.00

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