Question

Suppose the weight of pieces of passenger luggage for domestic airline flights follows a normal distribution with μ = 26 pounds and σ = 6.7 pounds.

(a) Calculate the probability that a piece of luggage weighs less than 31.2 pounds. (Assume that the minimum weight for a piece of luggage is 0 pounds.)


(b) Calculate the weight where the probability density function for the weight of passenger luggage is increasing most rapidly.
lb

(c) Use the Empirical Rule to estimate the percentage of bags that weigh more than 12.6 pounds.
%

(d) Use the Empirical Rule to estimate the percentage of bags that weigh between 19.3 and 39.4.
%

(e) According to the Empirical Rule, about 84% of bags weigh less than pounds.

Answer 1

Mean = 26 lb and standard deviation = 6.7 lb

(a)

• The z-score for 31.2 lb :
  Σ –μ 31.2 - 26 – = 0.7761 6,7
  
• From the standard normal distribution table: P(Z<0.7761) = 0.7811

(b)

• The probability density function of the standard normal distribution is
  = (r)
  
• The slope of the function or the rate of change is given by:
  f'(2) = J2T ?
  
• The point where the slope is maximum(i.e., rate maximum) is the point where the probability density function increases rapidly.
• To find max:
• Taking the derivative of the function
  (f":1) = 0)
  and equating to zero, we get the solution as x=1(minima) and x= -1(maxima)
• i.e., The Probability density function increases rapidly at the standard deviation -1 and decreases rapidly at standard deviation 1.
• So, the weight at which the function increases most rapidly is 26 - 6.7 = 19.3 lb

(c)

  

34% 34% 2.35% 13.5% ο 35% μ- 3σ μ- 2σ μ- Ισ μ μ+1σ μ+2σ μ+ 3σ

• 12.6 lb is 2 standard deviations left to the mean of the normal distribution.
• According to Empirical rule, 95% values fall within 2 standard deviations.
• So, the % of bags less than 2 standard deviations + % of bags greater than 2 standard deviations= 5%
• Therefore, % of bags weighings less than 12.6 = 5/2=2.5%
• So, % of bags weighing more than 12.6 lb is 100 - 2.5 = 97.5%

(d)

• 19.3 is one standard deviation left to the mean and 39.4 is 2 standard deviation right to the mean.
• According to Empirical rule, 68% values lie between one standard deviation and 95% values lie within 2 standard deviations of the standard normal distribution curve.
• This implies that 95-68=27% lies between standard deviations 1 and 2 on both sides, i.e., 13.5% values lie between standard deviation 1 and 2 on one side.
• So, the percentage of bags between 19.3 and 39.4 = % of bags within one standard deviation + % of bags between 1 and 2 standard deviations on one side
• i.e., % of bags between 19.3 and 39.4 = 65% + 13.5% = 78.5%  

(e)

• Acoording to Empirical rule 68% lies within 1 standard deviation or 34% on either side of the mean within one standard deviation.
• The normal bell curve is symmetrical and 50% lies on either side.
• So, 84% implies that it includes the left half of the curve and a standard deviation to the right of the mean(50+34).
• i.e., 84% of the bags weigh less than 26 + 6.7 = 32.7 lb