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Nancy Pfenning
nancyp+@pitt.edu
  

Stat 800 Normal Exercise Solutions

  1. The mean is 100 points, standard deviation is 10 points.
    1. Find the proportion of values below 115 points. The z score is (115-100)/10=1.5. Percentage below z=+1.5 is 93%.
    2. The lowest 84% of the values are below what value? The corresponding z score is +1.00, so the unstandardized value is 100+1.00(10)=110.
  2. The mean is 6 inches, standard deviation is 1.5 inches.
    1. Find the percentage of values below 5 inches. The z score is (5-6)/1.5=-.67. Percentage below -.67 is 25%.
    2. The tallest 16% are taller than how many inches? 16% above is the same as 84% below: z=+1.00, unstandardized value is 6+1(1.5)=7.5.
  3. The mean is 20 cigarettes, standard deviation is 5 cigarettes.
    1. Find the percentage of values above 22.5 cigarettes. z=(22.5-20)/5 =.5. The percentage above +.5 = percentage below -.5 = 31%.
    2. The top 90% are greater than how many cigarettes? 90% above means 10% are below. 10% corresponds to z=-1.28. Unstandardized, we have 20-1.28(5) = 13.6.
  4. The mean is 165 lbs., standard deviation is 12 lbs.
    1. Find the percentage of values above 141 lbs. z-value is (141-165)/12 = -2. The percentage of values above z=-2 is the same as the percentage below z=+2, which is .975.
    2. The lightest 2% are lighter than how many pounds? 2% below corresponds to z= -2.05. The unstandardized value is 165 - 2.05(12) = 140.4.
  5. The mean is 4 people, standard deviation is 1.3 people. Find the percentage of values between 2 and 6 people. z = (2-4)/1.3 = -1.54 and z=(6-4)/1.3=+1.54; the percentage between is 94%-6%=88%.
  6. The mean is 11 years, standard deviation is 2 years. Find the percentage between 16 and 17 years. z =(16-11)/2=2.5 and z= (17-11)/2 = 3; the percentage between is .9987-.995=.0037.
  7. The mean is $30,000; standard deviation is $8,000. Find the percentage of values between $20,000 and $22,000. The z-values are -1.25 and -1.00, corresponding to percentages .11 and .16. Subtracting, we get .05.
  8. The mean is 300 ml., standard deviation is 3 ml. Find the percentage of values
    1. less than 280 ml. This is almost 7 standard deviations below; practically none (percentage approx. 0%) are below this.
    2. greater than 280 ml. Practically all (percentage approx. 100%) are above.
    3. less than 315 ml. This is 5 standard deviations above; practically all (percentage approx. 100%) are below this
    4. greater than 315 ml. Practically none (percentage approx. 0%) are above this.

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