Transcript Statistics 101

Statistics 101
The Binomial Distributions
The Binomial Setting
• Each observation falls into one of two
categories (success or failure)
• Fixed n
• The observations are all independent
• The probability of success, p, is the same
for each observation
Binomial Distribution
• The distribution of the count X of
successes in a binomial setting is the
binomial distribution with parameters n
and p.
• n is the number of observations
• p is the probability of success
• X is B(n,p)
Examples
• Blood type is inherited. If both parents
carry genes for the O and A blood types,
each child has a probability 0.25 of getting
two O genes. Different children inherit
independently. The number of O blood
types among 5 children of these parents is
the count X of successes in 5 independent
observations with probability 0.25. So X
has the binomial distribution B(5, 0.25)
Example
• Deal 10 cards from a shuffled deck and
count the number X of red cards. Success
is a red card. Is this a binomial
distribution?
• No, Because the observations are not
independent therefore, it is not a binomial
distribution.
Try exercise 8.1 on pg 441.
• (a) No: There is no fixed n (i.e., there is no definite upper
limit on the number of defects).
• (b) Yes: It is reasonable to believe that all responses are
independent (ignoring any “peer pressure”), and all have
the same probability of saying “yes” since they are
randomly chosen from the population. Also, a “large city”
will have a population over 1000 (10 times as big as the
sample).
• (c) Yes: In a “Pick 3” game, Joe’s chance of winning the
lottery is the same every week, so assuming that a year
consists of 52 weeks (observations), this would be
binomial.
Example 8.5
Inspecting Switches
A quality engineer selects an SRS of 10 switches from a
large shipment for detailed inspection. Unknown to the
engineer, 10% of the switches in the shipment fail to
meet the specifications. What is the probability that no
more than 1 of the 10 switches in the sample fail
inspection?
B(10, 0.1)
Finding binomial probabilities
Probability histogram for the binomial distribution with n=10 and
p=0.1
Calculations
• We want to calculate
P(X< 1) = P(x=0) + P(x = 1)
• TI-83 command binompdf(n,p,X)
• 2nd (Distri)/0: binompdf
• (10, .1, 0) returns .3486784401
• (10, .1, 1) returns 0.387420489
• Sum and we get 0.7361
• Or about 74% of all samples will contain no
more than 1 bad switch
Cumulative distribution function
(cdf)
• Calculates the sum of the probabilities for
0, 1, 2, up to the value of X.
• For the count X of defective switches
previously done
•
binomcdf (10, .1, 1) returns
0.736098903