STATS Module 6

• Continuous random variable


  A variable that can assume any value on a continuum (can assume an uncountable number of values)

  
• Examples of continuous variables

  • Thickness of an item
  • Time required to complete a task
  • Temperature of a solution
  • Height, in inches
• Continuous probability distribution


  • The probability that x assumes a value in any interval lies in the range 0 to 1
  • The total probability of all the (mutually exclusive) intervals within which x can assume a value is 1.0

  
• The total area under the probability distribution curve of a continuous random variable is always 1.0, or 100%

  
• The area under the probability distribution curve of a continuous random variable between any two points is between 0 and 1

  
• Probability that a continuous random variable x assumes a value within a certain interval


  Given by the area under the curve between the two limits of the interval

  
• The probability that a continuous random variable x assumes a single value is always zero
• Normal probability distribution


  A bell-shaped (symmetric) curve with mean μ and standard deviation σ

  
• Characteristics of a normal probability distribution


  • The total area under the curve is 1.0
  • The curve is symmetric about the mean
  • The two tails of the curve extend indefinitely

  
• Approximately 99.7% of the observations in a normal distribution lie within three standard deviations of the mean
• Standard normal distribution


  The normal distribution with μ = 0 and σ = 1

  
• z Values or z Scores


  The units marked on the horizontal axis of the standard normal curve, representing the distance between the mean and a point in terms of standard deviations

  
• Converting an x Value to a z Value

  z = (x - μ) / σ
• The shape of the distribution is the same when converting between x and z units, only the scale has changed
• z


  Standardized normal random variable
• Probabilities for the standard normal curve


  • P(0 < z < 5.67)
  • P(z < -5.35)

  
• Converting an x Value to a z Value

  1. z = (x - μ) / σ

  2. where μ and σ are the mean and standard deviation of the normal distribution of x, respectively

  
• The shape of the distribution is the same, only the scale has changed when comparing X and Z units

  
• Finding Normal Probabilities


  1. Convert X to Z

  2. Calculate P(Z < 0.12)

  3. P(X < 18.6) = P(Z < 0.12)
• Finding Normal Upper Tail Probabilities


  1. Convert X to Z

  2. Calculate P(Z > 0.12)

  3. P(X > 18.6) = P(Z > 0.12) = 1.0 - P(Z ≤ 0.12)

  
• Finding a Normal Probability Between Two Values


  1. Convert X to Z

  2. Calculate P(0 < Z < 0.12)

  3. P(18 < X < 18.6) = P(0 < Z < 0.12) = P(Z < 0.12) - P(Z ≤ 0)

  
• Finding an x Value for a Normal Distribution

  x = μ + z σ

  
• z


  Standard score
• z = (x - μ) / σ
• Example 6-18


  • Life of a calculator manufactured by Calculators Corporation has a normal distribution with a mean of 54 months and a standard deviation of 8 months
• Warranty period


  Period to replace a malfunctioning calculator so the company does not want to replace more than 1% of all the calculators sold
• Calculating warranty period


  x = μ + z σ = 54 + (-2.33)(8) = 35.36 months
• The normal distribution is used as an approximation to the binomial distribution when np and nq are both greater than 5

  
• Binomial distribution


  • Applied to a discrete random variable
  • Each repetition results in one of two possible outcomes (success or failure)
  • Probabilities of the two outcomes remain the same for each repetition
  • Trials are independent
• Binomial formula


  Gives the probability of x successes in n trials
• Example 6-20


  • 50% of people in the US have at least one credit card, what is the probability that 19 out of 30 randomly selected people have at least one credit card
• Continuity correction factor is the addition or subtraction of 0.5 when using the normal distribution to approximate the binomial distribution
• Example 6-21


  • 32% of people working from home said the biggest advantage is no commute, what is the probability that 108 to 122 out of 400 randomly selected people will say this
• Example 6-22


  • 61% of women support red light cameras, what is the probability that 500 or more out of 800 randomly selected women will support them
• The probability that exactly 7 out of 10 people used an online travel website is 0.27
• The probability that at least 9 out of 10 people used an online travel website is 0.18