Exam 3 Review

Chapter 10

• Binomial Random Variable

  • Binomial Experiment

    • A fixed number (n) trials

    • Each trial must be independent of the others

    • Each trial has just two possible outcomes, called success (the outcome of interest) and failure

    • There is a constant probability (p) of success for each trial, the complement of which is the probability (1-p) of failure

  • Rule of Thumb

    • The number (X) of successes in a sample of size n taken without replacement from a population with proportion (p) of successes is approximately binomial with n and p as long as the sample size (n) is at most 10% of the population size (N)

    • n <= .10N or N >= 10n

  • The probability of exactly x successes (and n-x failures) in n trials

    • P(X=x) = n!/x!(n-x)! * p^x (1-p)^(n-x)

      • where x may take any value 0, 1, …, n

    • Excel Commands:

      • P(X=x): the probability of getting exactly x successes out of n trials

        • =BINOM.DIST(x,n,p,0) or BINOM.DIST(x,n,p,FALSE)

      • P(X<=x): the probability of getting at most x successes out of n trials

        • =BINOM.DIST(x,n,p,1) or BINOM.DIST(x,n,p,TRUE)

      • P(X>=x): the probability of getting at least x successes out of n trials

        • =BINOM.DIST(x-1,n,p,1) or BINOM.DIST(x-1,n,p,TRUE)

• Standardizing Values

  • z-score = (value - mean)/standard deviation

    • the convention is to denote a value of our normal random variable X with the letter “x.” since the mean is written as mu and the standard deviation sigma, we may write the standardized value as z-score

    • the standard value z tells how many standard deviations below or above the mean the original value is

    • formulas:

      • z = x- mu/sigma

      • EXCEL: =STANDARDIZE(x,mu,sigma)

      • finding x-values given z-scores:

        • x = mu + z * sigma

  • Standard Normal Distribution

    • P(z < -2.21); EXCEL: =NORM.S.DIST(-2.21,1)

    • P(z > 0.75); EXCEL: =1-NORM.S.DIST(0.75,1)

    • P(-1 < z < +1); EXCEL: =NORM.S.DIST(1,1)-NORM.S.DIST(-1,1)

    • (mean=0, standard deviation=1)

      • To find the z-value with an area of 0.01 below, it: =NORM.S.INV(0.01)

      • To find the z-value with an area of 0.15 above it: =NORM.S.INV(1-0.15)

  • Non-Standard Normal Values

    • Area Under Normal Curve:

      • P(X < x); EXCEL: =NORM.DIST(x, mu, sigma,1)

      • P(X > x); EXCEL: 1-NORM.DIST(x, mu, sigma, 1)

      • P(a < X < b); EXCEL: =NORM.DIST(b, mu, sigma, 1) - NORM.DIST(a, mu, sigma, 1)

  • Normal Approximation to the Binomial

    • Rule of Thumb

      • Probabilities for a binomial random variable X with n and p may be approximated by those for a normal random variable having the same mean and standard deviation as long as the sample size n is larger enough relative to the proportions of successes and failures, p and 1-p

        • np>=10 and n(1-p)>=10