381 Continuous Probability Distributions (The Normal Distribution-II) QSCI 381 – Lecture 17 (Larson and Farber, Sect )

381 Finding z-scores-I Yesterday we addressed the question: What is the probability that a normal random variable, X, would lie between x 1 and x 2. To address this question we found the probabilities P[X  x 1 ] and P[X  x 2 ] and calculated the difference between them. Today we are going to address the inverse of this question. Find the z-score which corresponds to a cumulative area under the standard normal curve of p.

381 Finding z-scores-II What value of x corresponds to an area of 0.8? X?X? Area=0.8

381 Finding z-scores-II We can use a table of z-scores or the EXCEL function NORMINV: NORMINV(p, ,  ) Once you have a z-score for a given cumulative probability, you can find x for any  and  using the formula:

381 Example-I The length distribution of the catch of a given species is normally distributed with mean 500 mm and standard deviation 30 mm. Find the maximum length of the smallest 5%, 50% and 75% of the catch.

381 Example-II Find the z-score for each level (5%, 50% and 75%)

381 Example-III We now apply the formula: so the maximum lengths are 450.7, 500 and mm.

381 Sampling and Sampling Distributions-I So far we have been working on the assumption that we know the values for  and . This is rarely the case and generally we need to estimate these quantities from a sample. The relationship between the population mean and the mean of a sample taken from the population is therefore of interest.

381 Sampling and Sampling Distributions-II A is the probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population. If the sample statistic is the sample mean, then the distribution is the sampling distribution of sample means.

381 Sampling and Sampling Distributions-III (Example) Consider a population of fish in a lake. The mean and standard deviation of the lengths of these fish are 300 mm and 50 mm respectively. We now take 100 random samples where each sample is of size 10, 20, or 100. What can we learn about the population mean?

381 Sampling and Sampling Distributions-IV (Example) N=10 N=20 N=100

381 Properties of the Sampling Distribution for the Sample Mean The mean of the sample means is equal to the population mean: The standard deviation of the sample means is equal to the population standard deviation divided by the square root of n. is often called the

381 The Central Limit Theorem 1. If samples of size n (where n  30) are drawn from any population with a mean  and a standard deviation , the sampling distribution of sample means approximates a normal distribution. The greater the sample size, the better the approximation. 2. If the population is itself normally distributed, the sampling distribution of the sample means is normally distributed for any sample size.

381 The Central Limit Theorem (Example)

381 Probabilities and the Central Limit Theorem The distribution of the heights of trees are not normally distributed. We sample 100 (of many) trees in a (very large) stand and calculate sample mean and sample standard deviation to be 12.5m and 2.3m respectively. What is the standard deviation of the mean? What is the probability that the population mean is less than 12m?