Example. Suppose that lengths of tails of adult Ring-tailed Lemurs are normally distributed with mean 50 cm and standard deviation 5 cm.

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1 Example Suppose that lengths of tails of adult Ring-tailed Lemurs are normally distributed with mean 50 cm and standard deviation 5 cm.

2 Example Suppose that lengths of tails of adult Ringtailed Lemurs are N(50 cm, 5 cm). (a) What is the probability that a randomly selected adult ring-tailed lemur has a tail that is 45 inches cm or shorter?

3 Example Suppose that lengths of tails of adult Ringtailed Lemurs are N(50 cm, 5 cm). (b) What is the probability that a randomly selected adult lemur has a tail that is 55 cm or longer?

4 Example Suppose that lengths of tails of adult Ringtailed Lemurs are N(50 cm, 5 cm). (c) Complete the sentence. Only 10% of all adult ring-tailed lemur population have a tail that is cm or longer?

5 Example Suppose that lengths of tails of adult Ringtailed Lemurs are N(50 cm, 5 cm). (d) Two adult ring-tailed lemurs will be randomly selected. What is the probability that both lemurs will have a tail that is 55 cm or longer?

6 Binomial Random Variables There are n trials (performances of the binomial experiment), where n is determined, not random. Each trial results in only 1 outcome: success or failure. The probability of a success on each trial is constant, denoted p, with 0 p 1. The probability of a failure q=1-p, with 0 q 1. The trials are independent.

7 Binomial Distribution X~ Binomial (n, p) Probability of exactly k successes in n trials: where Mean (Expected value) of X is Standard Deviation of X is

8 Example: Antibiotic Antibiotic with p=0.7. Suppose an antibiotic has been shown to be 70% effective against common bacteria. (a) If the antibiotic is given to 5 unrelated individuals with the bacteria, what is the probability that it will be effective for all 5? X = # ind. antibiotic is effective X ~ Binomial (5, 0.7)

9 Example: Antibiotic Antibiotic with p=0.7. Suppose an antibiotic has been shown to be 70% effective against common bacteria. (b) If the antibiotic is given to 5 unrelated individuals with the bacteria, what is the probability that it will be effective for the majority?

10 Example: Antibiotic Antibiotic with p=0.7. Suppose an antibiotic has been shown to be 70% effective against common bacteria. (c) If the antibiotic is given to 50 unrelated individuals with the bacteria, what is the probability that it will be effective for the majority, i.e., here it means for 26 or more individuals? Question: Computation of 25 sums is difficult, is there a better strategy?

11 Approximating Binomial Distribution Probabilities

12 Approximating Binomial Distribution Probabilities If the distribution of the number of successes is Binomial with the large number of trials n and the probability of a single success p not close to 0 or 1, the normal distribution can be used to approximate binomial probabilities. In general, the normal approximation is appropriate if KEY: convert X~ Binomial(n, p) into X~ N(μ, σ) and then into Z ~ N(0,1):

13 Example; Antibiotic Antibiotic with p=0.7. Suppose an antibiotic has been shown to be 70% effective against common bacteria. (c) If the antibiotic is given to 50 unrelated individuals with the bacteria, what is the probability that it will be effective for the majority? Step 1: Check: Step 2: Compute: Step 3: Calculate P(X>=50) assuming X~ (approx)n(μ, σ)

14 Example: Antibiotic Antibiotic with p=0.7. Suppose an antibiotic has been shown to be 70% effective against common bacteria. (c) If the antibiotic is given to 50 unrelated individuals with the bacteria, what is the probability that it will be effective for the majority, i.e., here it means for 26 or more individuals? np n(1 50*0.7 p) np 50*0.3 35, np(1 5 p) 3.24

15 Example: Antibiotic Antibiotic with p=0.7. Suppose an antibiotic has been shown to be 70% effective against common bacteria. (c) If the antibiotic is given to 50 unrelated individuals with the bacteria, what is the probability that it will be effective for the majority, i.e., here it means for 26 or more individuals? P( X 1 np 26) P( Z 35, 1 P( X 2.78) np(1 1 p) 26) P( Z )

16 Birthday Example Your close friends are making a surprise party for your birthday. They have already invited 100 people. Each person will accept the invitation with probability p =0.7 What is the probability that at least a half will show up? P(X>=50) =? np 70 5 n(1 p) 30 5 np 70, np(1 p) 4.58

17 Birthday Example Your close friends are making a surprise party for your birthday. They have already invited 100 people. Each person will accept the invitation with probability p =0.7 What is the probability that at least a half will show up? P(X>=50) =? np 70, np(1 p) 4.58 P( X 50) 1 P( X 50) 1 P( Z ) P( Z 4.37) 1 0 1

18 Population Parameter vs Statistic True population parameter value is usually unknown KYE IDEA: Take a sample and use the sample statistic to estimate the parameter. The sample statistic estimates (NOT necessarily equal to) a population parameter; in fact, it could change every time we take a new sample.

19 Population Parameter vs Statistic Question: Will the observed sample statistic value be a reasonable estimate? Answer: If our sample is a random sample and the size of sample is sufficiently large, then we will be able to say something about the accuracy of the estimation process.

20 Sampling Distribution Sampling distribution of statistic - distribution of all possible values of a statistic for repeated samples (same size) from target population. With a large number of samples, can assess if statistic will be close and how close on average to the true population parameter. Many of the sample statistics (e.g., sample mean, sample proportion, etc.) have approximately normal distributions

21 Estimating true population parameter When estimating the true population parameter with the sample statistics we would like: sampling distribution to be centered at the true parameter (unbiased statistic) variability in the estimates to be as small as possible

22 Estimating true population parameter Sampling distribution applet:

23 Sampling Distribution of Sample Mean The sampling distribution of the Sample Mean, X, is the distribution of the sample mean values for all possible samples of the same size from the same population. If the parent population IS N(μ, σ), then for any sample size (small or large), the distribution:

24 Central Limit Theorem (CLT) The sampling distribution of the Sample Mean,, is the distribution X of the sample mean values for all possible samples of the same size from the same population. If the parent population IS NOT N(μ, σ), then for n>30 the distribution is approximately:

25 Example: Research on eating disorder Bulimia is an illness in which a person binges on food or has regular episodes of significant overeating and feels a loss of control. Usually is observed in young females.

26 Research on eating disorder During the American Statistician (May 2001) study of female students who suffer from bulimia, each student completed a questionnaire from which a fear of negative evaluation (FNE) score was produced. Suppose the FNE scores of bulimic students have a distribution with mean 18 and standard deviation of 5. What is the probability that a randomly selected bulimic female has a greater than 15?

27 Research on eating disorder During the American Statistician (May 2001) study of female students who suffer from bulimia, each student completed a questionnaire from which a fear of negative evaluation (FNE) score was produced. Suppose the FNE scores of bulimic students have a distribution with mean 18 and standard deviation of 5. What is the probability that the sample mean FNE score (of 100) is greater than 15?

28 Research on eating disorder During the American Statistician (May 2001) study of female students who suffer from bulimia, each student completed a questionnaire from which a fear of negative evaluation (FNE) score was produced. Suppose the FNE scores of bulimic students have a distribution with mean 18 and standard deviation of 5. What is the probability that the sample mean FNE score (of 25) is greater than 15?

29 Example: Body fat of American Men The percentage of fat (not the same as body mass index) in the bodies of American men is an approx. normal with mean of 15% and std of 5%. If these values were used to describe the body fat of U.S. Army men, then a measure of 20% or more body fat would characterize a soldier as obese.

30 Body fat of American Men The percentage of fat of American men is an approx. normal (mean=15, std =5) Measure of 20% or more body fat would characterize a U.S. Army soldier as obese. What is the probability that a randomly chosen soldier would be considered obese, i.e., the percentage of body fat of a randomly chosen soldier in the U.S. Army is 20% or higher?

31 Body fat of American Men The percentage of fat of American men is an approx. normal (mean=15, std =5) Measure of 20% or more body fat would characterize a U.S. Army soldier as obese. What is the probability that an average percentage of body fat of a randomly chosen 100 soldiers would be 20% or higher?

32 Body fat of American Men The percentage of fat of American men is an approx. normal (mean=15, std =5) Measure of 20% or more body fat would characterize a U.S. Army soldier as obese. Given that the distribution of the percentage of the body fat of U.S. Army soldiers has the mean of 13% and the standard deviation of 3% (not necessarily normal), what is the probability that an average percentage of body fat of a randomly chosen 100 soldiers would be 20% or higher?