1. Copyright ©2011 Nelson Education Limited The Normal Probability Distribution CHAPTER 6

2. Copyright ©2011 Nelson Education Limited Properties of Continuous Probability Distributions 1.The area under the curve is equal to 1. area under the curveP(a  x  b) = area under the curve between a and b. There is no probability attached to any single value of x. That is, P(x = a) = 0.

3. Copyright ©2011 Nelson Education Limited Example Suppose that a random variable X is uniformly distributed on the interval [0,3]. That is, its density is some constant c on [0,3], and is zero otherwise. What is the value of c?

4. Copyright ©2011 Nelson Education Limited The Normal Distribution The shape and location of the normal curve changes as the mean and standard deviation change. The formula that generates the normal probability distribution is: APPLET MY

5. Copyright ©2011 Nelson Education Limited Using standard normal Table The four digit probability in a particular row and column of Table 3 gives the area under the z curve to the left that particular value of z. Area for z = 1.36

6. Copyright ©2011 Nelson Education Limited Finding Probabilities for the General Normal Random Variable To find an area for a normal random variable x with mean  and standard deviation  standardize or rescale the interval in terms of z. Find the appropriate area using Table 3. To find an area for a normal random variable x with mean  and standard deviation  standardize or rescale the interval in terms of z. Find the appropriate area using Table 3. Example: Example: x has a normal distribution with  = 5 and  = 2. Find P(x > 7). 1 z

7. Copyright ©2011 Nelson Education LimitedExample The weights of packages of ground beef are normally distributed with mean 1Kg and standard deviation 0.15Kg. What is the probability that a randomly selected package weighs between 0.7 and 0.775 Kgs? What is the weight of a package such that only 1% of all packages exceed this weight?

8. Copyright ©2011 Nelson Education Limited Sampling Distributions CHAPTER 7

9. Copyright ©2011 Nelson Education LimitedIntroduction Parameters are numerical descriptive measures for populations. –For the normal distribution, the location and shape are described by  and  –For a binomial distribution consisting of n trials, the location and shape are determined by p. Often the values of parameters that specify the exact form of a distribution are unknown. You must rely on the sample to learn about these parameters.

10. Copyright ©2011 Nelson Education Limited Sampling Distributions statistics Numerical descriptive measures calculated from the sample are called statistics. Statistics vary from sample to sample and hence are random variables. sampling distributions The probability distributions for statistics are called sampling distributions. In repeated sampling, they tell us what values of the statistics can occur and how often each value occurs.

11. Copyright ©2011 Nelson Education Limited Example: A population is made up of the numbers 3,5,2,1. You draw a sample of size n=3 without replacement and calculate the sample average. Find the distribution of Xbar, the sample average.

12. Copyright ©2011 Nelson Education Limited If random samples of n observations are drawn from any population with finite  and standard deviation , then, when n is large, the sampling distribution of is approximately normal, with mean  and standard deviation. The approximation becomes more accurate as n becomes large. If random samples of n observations are drawn from any population with finite  and standard deviation , then, when n is large, the sampling distribution of is approximately normal, with mean  and standard deviation. The approximation becomes more accurate as n becomes large. Central Limit Theorem 1.any population 2.random sampling (independence) 3.if start with normal, this is exact! 4.how large does n have to be?

13. Copyright ©2011 Nelson Education Limited Many (most?) statistics used are sums or averages. The CLT gives us their sampling distributions. Overall, we have that The average of n measurements is approximately normal with mean  and variance σ 2 /n. The sum of n measurements is approximately normal with mean n  and variance nσ 2. The sample proportion is approximately normal with mean p and variance p(1-p)/n. The binomial is approximately normal with mean np and variance np(1-p).

14. Copyright ©2011 Nelson Education Limited How Large does n need to be? If the original distribution is normal, then any sample size will do. If the sample distribution is approximately normal, then even small n will work. If the sample distribution is skewed, then a larger n is needed (eg. n bigger than 30).

15. Copyright ©2011 Nelson Education Limited The CLT in the real-world: 1.Check the random sampling – was it done properly? Are your observations independent? 2.Look at a histogram of your data to see if it is skewed or symmetric. The more skewed the data, the less credible the approximation. Is your sample size big enough? Rules of thumb: a.In general, use n≥30 for skewed distributions b.Use np, n(1-p)≥ 5 for the binomial/sample proportion.

16. Copyright ©2011 Nelson Education Limited Example Suppose X is a binomial random variable with n = 30 and p =.4. Use the normal approximation to find P(X  10).

17. Copyright ©2011 Nelson Education LimitedExample A production line produces AA batteries with a reliability rate of 95%. A sample of n = 200 batteries is selected. Find the probability that at least 195 of the batteries work.

18. Copyright ©2011 Nelson Education LimitedExample A bottler of soft drinks packages cans in six-packs. Suppose that the fill per can has an approximate normal distribution with a mean of 355 ml and a standard deviation of 5.91 ml. What is the probability that the total fill for a case is less than 347.79 ml?

19. Copyright ©2011 Nelson Education Limited Example The soda bottler in the previous example claims that only 5% of the soda cans are under filled. A quality control technician randomly samples 200 cans of soda. What is the probability that more than 10% of the cans are under filled?