8'1 Day 2: Binomial Distributions

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8.1 Day 2 Binomial Distributions
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Binomial Mean Standard Deviation
Beware!!! These short formulas are good only for
binomial distributions. They cant be used for
other discrete random variables.

• If a count X has a binomial distribution, the
  mean and standard deviation are

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Ex 1 Inspecting Part 3

• The count X of bad switches is binomial with n
  10 and p 0.1. This is the sampling
  distribution the engineer would see if she drew
  all possible SRSs of 10 switches from the
  shipment and recorded the value of X for each
  sample.

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Normal Approximation to Binomial Distributions

• As the number of trials gets larger, the binomial
  distribution approaches a Normal distribution.
  We may then use Normal probability calculations
  to approximate binomial probabilities.

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Ex 2 Shopping

• Sample surveys show that fewer people enjoy
  shopping than in the past. A survey asked a
  nationwide random sample of 2500 adults if they
  agreed or disagreed that I like buying new
  clothes, but shopping is often frustrating and
  time-consuming. The population that the poll
  wants to draw conclusions about is all U.S.
  residents aged 18 and over. Suppose that in fact
  60 of all adult U.S. residents would say Agree
  if asked the same question. What is the
  probability that 1520 or more of the sample agree?

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Ex 2 Shopping
As a rule of thumbwe will use the Normal
approximation when n and p satisfy npgt10 and n(1
- p)gt10

• The find the probability that at least 1520 of
  the people in the sample find shopping
  frustrating, we must add the binomial
  probabilities of all outcomes from X 1520 to X
  2500. This isnt practical! Instead

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Ex 3 Free Throws - SIMULATION

• Yesterday, we calculated that the binomial
  probability (P(Xlt7)) that Kevin Garnett made at
  most 7 out of 12 shots was 0.1576.
• To simulate in the calculator
• Let X number of makes in 12 attempts. (Note
  that the probability of success was 0.75)
• We will assign 0 to a miss and 1 to a make.
• randBin(1,0.75,12) simulates 12 attempts.
• You Try

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Ex 3 Free Throws - SIMULATION

• One way to run many simulations
• randBin(1,0.75,12) L1sum(L1)
• Continue pressing the ENTER key until you have 10
  numbers. Record these numbers. Calculate the
  relative frequency (or proportion of Xlt7).
• Press ENTER until you have 20 numbers. Calculate
  the relative frequency.
• Press ENTER until you have 50 numbers. Calculate
  the relative frequency. What do you notice?