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NORMAL DISTRIBUTION INVERSE NORMAL Z-CURVE BINOMIAL DISTRIBUTION NORMAL APPROXIMATION TO BINOMIAL Chapter 4 Binomial and Normal Distributions

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Normal Distribution: N(mean,var) Symmetric about the mean The mean divides the curve in 2 halves The probability is the area under the curve The total area or probability under the curve is 1 It is completely determined by the mean and standard deviation

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Normal Distribution They sell an average of 36 TVs with standard deviation of 8. How many TVs do they need to have to meet the demand? Compute the probability of selling 40 or more TV sets. Histogram and stemplot of sales data of large screen TVs.

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Normal Distribution Computing Probabilities using TI 2 nd Distribution Normalcdf Normalcdf(a,b,mean,standard deviation)

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Normal Distribution Sales of large screen TVs example Mean = 36 SD = 8 Compute the probability of selling 40 or more TV sets. P(X40) = Normalcdf(40,9999,36,8) = 0.3085

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Normal Distribution P(aXb)=normalcdf(a,b,mean,sd) e.g., Mean = 36 SD = 8 P(40X50)=normalcdf(40,50,36,8) =0.2685

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Normal Distribution P(Xa)=normalcdf(a,99999,mean,sd) e.g., Mean = 36 SD = 8 P(X40)=normalcdf(40,99999,36,8) =0.3085

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Normal Distribution P(Xa)=normalcdf(-99999, a,mean sd) e.g., Mean = 36 SD = 8 P(X40)=normalcdf(-99999, 40, 36, 8) =0.6915

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iClicker

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A study was conducted to determine if any link existed between cellular phone usage and brain cancer development. Data from this study indicated that the monthly phone usage for all users is approximately normally distributed with mean 2.4 hours and standard deviation 1.1 hours. What is the proportion of phone users who use their phones at least 3 hours a month? a. 0.7073 b. 0.2143 c. -0.2927 d. 0.6927 e. 0.2927

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Inverse Normal a = ? Such that P(Xa) = Percentile InvNorm(percentile, mean, sd) 2 nd distribution InvNorm Note: InvNorm computes a so that the area from the left most to a is the percentile

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Inverse Normal From the TV example, how many TVs do they need in stock to meet the demand? The store manager decided to order enough TVs to cover 90% of the demand. How many TVs should they order? a = number of TVs they should order P(Xa) = 0.90 a = InvNorm(.90,36,8) a = 46.25 46 TVs

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Inverse Normal Sales and Demand of TVs example Mean = 36 SD = 8 What is a when P(X a) = 0.70 a = InvNorm(.70, 36, 8) = 40.20 What is a when P(X a) = 0.30 a = InvNorma(.30, 36, 8) = 31.80 What is a when P(X a) = 0.60 P(X a) = 0.40 a = InvNorm(.40, 36, 8) = 33.97

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iClicker A study was conducted in order to determine if any link existed between cellular phone usage and the development of brain cancer. Data from this study indicated that the monthly phone usage for all users is approximately normally distributed with mean 2.4 hours and standard deviation 1.1 hours. Up to how many hours does the lower 80% of the users spend on the phone? a. 1.82 hours b. 3.33 hours c. 1.47 hours d. 3.12 hours e. 3.81 hours

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Z-Score Z is how many standard units above or below the mean (standard deviations from the mean) Z = (X – Mean)/ SD Properties of Z curve Z Curve is Normal with Mean = 0 and SD = 1 P ( -1 Z 1 ) =.68 P ( -2 Z 2 ) =.95 P ( -3 Z 3 ) =.997

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For a certain process, the chance of observing at most 10 cases is 5%. The process is known to have a standard deviation of 5. What is the mean of the process? Z-Score

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Binomial Distribution Properties of a Binomial Process trial performed n times and each trial is independent of each other each trial results in one of two possible outcomes (S = Success, F = Failure) the probability of success is p and the probability of failure is q = 1 – p for all trials Binomial Random Variable X = total number of successes Parameters: n = number of trials p = probability of success

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Binomial Distribution Example 1 A Stat 216 multiple choice quiz has 5 questions, with each question having 4 choices. a. Success = the question answered correctly b. X = number of correct answers c. X = {0,1,2,3,4,5} d. n = 5 e. p = ¼

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Binomial Distribution Example 2 Available data shows that 40% of telephone respondents agree to be interviewed for market research surveys. Suppose that the polling organization Reliable Research randomly selects and dials telephone numbers until 50 respondents are reached. a. Success = respondent agrees to be interviewed b. X = number of respondents who agree to be interviewed c. X = {0,1,2,3,…,49,50} d. n = 50 e. p = 0.40

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Binomial Distribution Calculator. P(X = j) = binompdf(n, p, j) 2 nd distribution binompdf Example1. Calculate the probability of each event occurring. (n=5, p=0.25) Note: The total probability should be 1.0 P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5) = 1

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iClicker

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Binomial Distribution Calculator: P(X j ) = binomcdf(n,p,j) P(X j ) = P(X = 0) + P(X = 1) +... + P(X = j) From Example1, what is the probability that a student will get at most 3 correct answers?

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iClicker

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Binomial Distribution From Example1, what is the probability that a student will get at least 2 correct answers?

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iClicker

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If X is defined to be a binomial random variable with parameters n and p then: the expected value or the mean of X is np the standard deviation of X is Binomial Distribution

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From Example2, how many respondents will be expected to agree to be interviewed? What is the corresponding mean and standard deviation? n = 50 p =.40 The researcher should expect to interview around 20 respondents give or take 4. Binomial Distribution

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Right-skewed Left-skewed Symmetric Normal Approximation to Binomial

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The normal curve gives reasonably good approximations of binomial probabilities whenever both np > 5 and n(1-p) > 5. To which case above will the normal approximation to the binomial be valid? npn(1-p) n = 30, p = 0.10327 n = 30, p = 0.8525.54.5 n = 30, p = 0.4513.516.5

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