Inverse Normal Distribution and Approximation to Binomial

Transcription

1 Inverse Normal Distribution and Approximation to Binomial Section 5.5 Cathy Poliak, Ph.D. Office in Fleming 11c Department of Mathematics University of Houston Lecture Cathy Poliak, Ph.D. Office in Fleming 11c Section (Department 5.5 of Mathematics University of Lecture Houston 16 ) / 23

2 Outline 1 Reveiw of Normal Distribution 2 Inverse Normal 3 Approximating the Binomial Distribution Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Section (Department 5.5 of Mathematics University of Lecture Houston 16 ) / 23

3 Popper 11 Questions Let a random variable X have a Normal distribution with mean µ = 10 and standard deviation σ = 2. For the following questions determine what is the proper way to solve these probabilities. 1. P(X < 7.25) 2. P(X 5) a) pnorm(7.25,10,2) c) pnorm(7,10,2) b) 1-pnorm(7.25,10,2) d) dnorm(7.25, 10, 2) a) pnorm(5, 10, 2) c) 1 - pnorm(4, 10, 2) b) 1 - pnorm(5, 10, 2) d) dnorm(6, 10, 2) 3. P(9 X 11) a) pnorm(11, 10, 2) - pnorm(8, 10, 2) b) pnorm(11, 10, 2) - 1- pnorm(9, 10, 2) c) pnorm(11, 10, 2) - pnorm(9, 10, 2) d) dnorm(11, 10, 2) - dnorm(9, 10, 2) Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Section (Department 5.5 of Mathematics University of Lecture Houston 16 ) / 23

4 Normal Distribution Calculations Area under a Normal curve represent proportions (probability) of observations within a range of values. There is no easy way to find the area under a Normal curve. We use a table or software that calculates the desired areas. The table we use is Z-table It uses a cumulative proportion. A cumulative proportion is the proportion (probability) of observations in a distribution that lie at or below a given value. This is Φ(z). When the distribution is given by a density curve, the cumulative proportion is the area under the curve to the left of a given value. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Section (Department 5.5 of Mathematics University of Lecture Houston 16 ) / 23

5 Using The Z-table The vertical margin are the left most digits of a z-score. The top margin is the hundredths place of a z-score. The numbers inside the table represents the area from to that z-score. Remember that the standard Normal density curve is symmetric and the total area is equal to 1. Note: R can calculate these probabilities and also some calculators. Without having to convert to z-scores. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Section (Department 5.5 of Mathematics University of Lecture Houston 16 ) / 23

6 Example Let X = amount of juice in ounces in a orange, X N(4.7, 0.4). 1. Determine the probability (using the z-table) that less than 5 ounces of juice are in an orange. 2. Determine the probability (using the z-table) that between 4 and 4.5 ounces of juice are in an orange. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Section (Department 5.5 of Mathematics University of Lecture Houston 16 ) / 23

7 Finding a value when given a proportion Called inverse Normal. This is working Backwards using Z-Table. Finding the observed values when given a percent. In R: qnorm(proportion,mean,sd). Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Section (Department 5.5 of Mathematics University of Lecture Houston 16 ) / 23

8 Backward Normal calculations Using Z-Table 1. State the problem. Since, Z-Table, qnorm and invnorm gives the areas to the left of z-scores, always state the problem in terms of the area to the left of x. Keep in mind that the total area under the standard Normal curve is Use Table A to find c. This is the value from the table not a value that we calculate. 3. Unstandardized to transform the solution from the z-score back to the original x scale. Solving for x using the equation gives the equation x = σ(c) + µ. c = x µ σ Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Section (Department 5.5 of Mathematics University of Lecture Houston 16 ) / 23

9 Examples to Work "Backwards" with the Normal Distribution Find the value of c so that: 1. P(Z < c) = P(Z > c) = P( c < Z < c) = Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Section (Department 5.5 of Mathematics University of Lecture Houston 16 ) / 23

10 MPG for Prius The miles per gallon for a Toyota Prius has a Normal distribution with mean µ = 49 mpg and standard deviation σ = 3.5 mpg. 25% of the Prius have a MPG of what value and lower? 1. We want c, such that P(Z < c) = That is we want to know what z-score cuts off the lowest 25%. P( Z <?) =0.25 z / 23

11 Find c such that P(Z < c) = From Table A, find something close to 0.25 inside the table. z P(Z <?) = 0.25 (closes value is ) z = (-0.6 row column ) / 23

12 Find c such that P(Z < c) = Unstandardized: x = σ(c) + µ = 3.5( 0.67) + 49 = This means that 25% of the Prius has a mpg of less than mpg. Using R: qnorm(0.25,49,3.5) = / 23

13 Top 10% Suppose you rank in the 10% of your class. If the mean GPA is 2.7 and the standard deviation is 0.59, what is your GPA? ( Assume a Normal distribution) 1. We want c, such that P(Z > c) = That is we want to know what z-score cuts off the highest 10%. P(Z >?) = 0.10 z / 23

14 Find c such that P(Z > c) = From Table A, the areas are below or to the left of a z-score thus we want to find something close to 0.90 inside the table. z P(Z <?) = 0.90 (close value is ) z = 1.28 (1.2 row column ) / 23

15 Find c such that P(Z > c) = Unstandardized: x = σ(c) + µ = 0.59(1.28) = This means that your gpa is if you rank at the 10% of your class. In R: qnorm(0.9,2.7,0.59) = / 23

16 Example Let X = amount of juice in ounces in a orange, X N(4.7, 0.4). 1. Determine the third quartile. 2. Determine the 95th percentile / 23

17 Approximation for Binomial Suppose a random variable X has a binomial distribution with p = 0.1. The following is a histogram with n = 10. n = 10, p= 0.1 Density / 23

18 Approximation for Binomial Suppose a random variable X has a binomial distribution with p = 0.1. The following is a histogram with n = 20. n = 20, p= 0.1 Density / 23

19 Approximation for Binomial Suppose a random variable X has a binomial distribution with p = 0.1. The following is a histogram with n = 50. n = 50, p= 0.1 Density / 23

20 Approximation for Binomial Suppose a random variable X has a binomial distribution with p = 0.1. The following is a histogram with n = 100. n = 100, p= 0.1 Density / 23

21 Theroem 5.3 Let X be a binomial random variable based on n trials with success probability p. Then if the binomial probability histogram is not too skewed, X has an approximate Normal distribution with µ = np and σ = np(1 p). In particular, for x = a possible value of X, P(X x) = Binom(x; n, p) (area under the normal curve to the left of x + 0.5) ( ) x np = Φ np(1 p) In practice, the approximate is adequate provided that both np 10 and n(1 p) / 23

22 Example of Normal Approximation Suppose that your mail-order company advertises that it ships 90% of its orders within three working days. Suppose you take a simple random sample of 100 orders: 1. What is the probability that 86 or fewer of the orders are shipped on time? 2. What is the probability that more than 95 of the orders are shipped on time? / 23

23 Popper 11 Questions 4. A 5. B / 23