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Mean and Standard Deviation Remember, to find the mean of a probability distribution, you take the sum of the probabilities times their value **sum of xP(x) The binomial distribution has special formulas for the mean and standard deviation Mean(µ) = np Standard Deviation(σ) = Where: n = # of observationsp = probability of success Beware!!! These formulas are BINOMIAL specific and don’t work for other distributions.

Mean and Standard Deviation A university claims that 80% of its basketball players get degrees. An investigation examines the fate of all 20 players who entered the program over a period of several years that ended six years ago. Of these players, 11 graduated and the 9 remaining are no longer in school. What is the standard deviation for the group of 30? How many athletes can the university expect to graduate out of their next crop of 30 athletes? Answer 30(.55) = 16.5 If the same group had a probability of success of.8, what would the standard deviation be? … p =.9? Notice that our σ gets smaller as our p value gets closer to 1.

Normal Approximation As the number of trials n, gets larger, a binomial probability distribution gets close to becoming a Normal distribution What does that have to do with us? Statisticians often use a “normal” approximation to find binomial probabilities for high sample spaces. How do use a normal curve to find probabilities? Change your X to Z and find the appropriate area!! This method was used to shorten large cumulative binomial functions before we had cdf’s in the calculator. While the calculator is better, we still need to know how and when to use the normal approximation. Why would we use this?

The “When” of the Normal Approximation In order to use the normal approximation, the following 2 conditions must be met: np ≥ 10 np ≥ 10 n(1-p) ≥ 10 n(1-p) ≥ 10 **Be sure to check these 2 conditions before you use a normal approximation Check the conditions and decide if a normal approximation would appropriate. ≥ (.91) = 368 ≥ (.09) = 32 ≥ 10 Normal Approximation is appropriate. Many local polls of public opinion use samples of size 400 to 800. Consider a poll of 400 adults in Richmond that asks the question, “Do you approve of President George W. Bush’s response to the World Trade Center terrorist attacks in September 2001?” Suppose we know the President’s approval rating on this issue nationally is 92%. You are asked to calculate the probability that at most 358 of the 400 adults in the Richmond Poll answer “yes” to the question.

The “How” of the Normal Approximation Here are the steps to a normal approximation: Find the µ and σ of the distribution. Find the µ and σ of the distribution. Change the X to a Z score Change the X to a Z score Use the z chart or your calculator to find the appropriate area under the curve. Use the z chart or your calculator to find the appropriate area under the curve.

Let’s Do It!! Many local polls of public opinion use samples of size 400 to 800. Consider a poll of 400 adults in Richmond that asks the question, “Do you approve of President George W. Bush’s response to the World Trade Center terrorist attacks in September 2001?” Suppose we know the President’s approval rating on this issue nationally is 92%. Use the Normal Approximation to calculate the probability of at most 358 people in the sample approving. µ = 400(.92) = 368 σ = sqrt(400*.92*.08) = X -> Z (358 – 368) / (5.4259) = Z-area for =.0329 Try getting the answer using your calculator and the Binomial CDF function. binomcdf(400,.92,358 =..0441

Note about the Normal Approximation Notice the difference between our approximate answer and the exact answer of the pdf. (.0441) – (.0329) =.0112 difference (.0441) – (.0329) =.0112 difference Not large, but not really good either!!! Accuracy of the Normal Approximation The Normal Approximation is MORE accurate when p is closer to ½. The Normal Approximation is LESS accurate when p is closer to 0 or 1.

Homework Read Pages 457,58 on Simulating Binomial Experiments Do Problem #’s 25,27-36 Bring today’s homework and what was Due today next class to be checked for a homework grade. We will check answers then as well.