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3. The Binomial Distribution for k = 0, 1, 2, …, n where which represents the number of ways to select k items from n

4. Try It! Finding Binomial Probabilities a.What is the probability of selecting exactly one shopper who actually makes a purchase? b.What is the probability of selecting exactly two shopper who actually makes a purchase? Random sample of n = 10 online shoppers from large population and p = 0.25 is the population proportion who actually make a purchase.

5. Try It! Finding Binomial Probabilities c.What is the probability of selecting at least one shopper who actually makes a purchase? Random sample of n = 10 online shoppers from large population and p = 0.25 is the population proportion who actually make a purchase. Try part (d) and click in your answer…

6. d.How many shoppers in your random sample of size 10 would you expect to actually make a purchase? A) 1 B) 2 C) 2.5 D) 3 E) Don’t know You just computed the mean of a binomial distribution!

7. Mean and Standard Deviation of a Binomial If X has the binomial distribution Bin(n,p), then Mean of X is  = E(X) = np Standard Deviation of X, is  =

8. Try It! More Work with the Binomial The mean or expected number of left-handed Americans in the sample is 12(0.10) = 1.2. The standard deviation would be  = =1.04. Suppose 10% of Americans are left-handed. Let X = the number of left-handed Americans in a random sample of size 12. Then X has a __________________________ distribution. a.What is the probability that the sample contains 2 or fewer left-handed Americans?

9. Try It! More Work with the Binomial b.Suppose a random sample of 120 Americans had been taken instead of just 12. So how might you find the probability that the sample would result in 20 or fewer left-handed Americans? Note: 2 out of 12 is 16.67% and 20 out of 120 is also 16.67%. Suppose 10% of Americans are left-handed. mean number of LH Americans in sample is 120(0.10) = 12. standard deviation would be  = =3.29.

10. 8.5 Continuous Random Variables Continuous random variable takes on all possible values in an interval. Probability = corresponding area under a curve called probability density function.

11. Density Curve Definition: A curve (function) is called a Probability Density Curve if: 1. It lies on or above the horizontal axis. 2. Total area under the curve is equal to 1. KEY IDEA: AREA under density curve over a range of values corresponds to PROBABILITY the random variable X takes on a value in that range.

12. Try It! Some Density Curves How would you use this curve to estimate the probability of a randomly selected employed adult from this city having an income $30,000 and $40,000? I. Density curve for modeling income (in $1000s) for a city.

13. a. Is this a density function? Try It! Some Density Curves II. Consider the following curve

14. Try It! Some Density Curves b. Find the probability of observing a response that is less than 35.

15. What does the value of 35 correspond to for this distribution? A) Q1 B) Median C) Q3

16. Try It! Checkout time at a store a.Draw the density. b. What is probability a person will take more than 10 minutes to check out? X = checkout time at a store, a random variable that is uniformly distributed on values between 5 and 20 minutes; that is, X is U(5,20) Try out the questions now!

17. Try It! Checkout time at a store c.Given already spent 10 minutes checking out, what is probability will take no more than 5 additional minutes? d. What is the expected time to check out at this store? X = checkout time is U(5,20) 1/15

18. Mean of a Continuous Random Variable Definition: Mean of a continuous random variable: Expected value or mean  = E(X) = balancing point (Sometimes need calculus/integration to find it -- integral instead of sums)

19. 8.6 Normal Random Variables Notation: The variable X is normally distributed with mean  and standard deviation  is denoted by:

20. Sketch Two Normal Curves N(50,10) and N(80,5)

21. Standard Score or z-score Finding Probabilities for z-scores: When we convert to standardized scores, the random variable X is converted to what is called the Standard Normal Random Variable, denoted by Z, and it has the N(0,1) distribution. Table A.1 provides the areas to the left for various values of Z

22. Table A.1: Area to left for various values of Z From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with permission.

23. Try It! Find Probabilities for Z 1.Find P(Z ≤ 1.22) Think about it: What is P(Z < 1.22)? 2. Find P(Z > 1.22) 3. Find P(-1.58 < Z < 2.24)

24. Try It! Find Probabilities for Z 4. What is probability a standard normal variable Z is within 2 standard deviations of mean? That is, find P(-2 ≤ Z ≤ 2). 5. What is P(Z ≤ 4.75)? 6. What is P(Z > 10.20)?

25. Try It! Find Probabilities for Z 7. What is the 90th percentile of the N(0,1) distribution? If the variable X has the N( ,  ) distribution, then the standardized variable, will have the N(0,1) distribution. How to find areas under other NORMAL distributions?