Probability Distributions for Continuous Random Variables

Transcription

1 1 Probability Distributions for Continuous Random Variables Section 5.1 A continuous random variable is one that has an infinite number of possible outcomes. Its possible values can form an interval. The probability distribution of a continuous random variable satisfies o Probabilities are areas under a density curve. o Total area under the density curve is exactly equal to one. o Each outcome has a zero probability of happening, but we can assign probabilities to intervals. We can sketch a smooth curve to help visualize the areas that are used to find the probability distribution of a continuous random variable. The smooth curve is almost a smoothed out histogram, representing an infinite number of people in a population. 1

2 2 Consider the family of Normal Distributions, which are o Bell-shaped and symmetric o Centered at their mean o With spread given by their standard deviation o Notation: X ~ N (, ) o Normal probability distribution function: f ( x) 1 2 e ( x 2 2 ) 2 For this family of distributions, we have been using the Empirical Rule to find probabilities. Now we will use the Standard Normal Table. Standard Normal Distribution A special case of the normal distribution in which the mean deviation =1. =o and the standard Notation: Z ~ N (0,1) Standard Normal Table The standard normal table is Table 4, pages A16-A17, in the appendix of the book. It gives areas under the normal curve to the left of the z-score; these are called cumulative probabilities. The z-scores appear on the margins of the table, areas are in the center. Remember, the area under the curve is the same thing as the probability of an event. 2

3 3 Section 5.2 Steps for Calculating Probabilities given x 1. Identify X, x,, and (or x, x, and s). 2. Convert X=x to Z = x 3. Look for the z-score along the margin of the Table to find the corresponding probability in the middle of the z-table. I find it helps to draw a picture: 3

4 4 Example The distribution of IQ scores is approximately normal with mean 100 and standard deviation 16. a) If Bubba has an IQ of 125, what s his z-score? b) What percentage of people IQs lower than 123? c) What percentage of people IQs lower than 89? 4

5 5 d) What percentage of people IQs higher than 89? e) What percentage of people IQs between 89 and 123? f) What percentage of people IQs of exactly 89? 5

6 6 Section 5.3 Steps for Calculating x given the Probability 1. Draw a picture, with the area given shaded on it. 2. Look up the cumulative area in the middle of the z-table, and look at the margins to find the z-score corresponding to that area. 3. Use the z-score formula to convert from Z = x to x = Z. 6

7 7 Example The distribution of IQ scores is approximately normal with mean 100 and standard deviation 16. a) What IQ corresponds to the bottom 20%? b) What probability corresponds to the top 5%? 7

8 8 Example You scored 650 on the SAT. Your friend took the ACT instead, and scored 30. The ACT that year had a mean of 21 and a standard deviation of 4.7. The SAT that year had a mean of 500 and a standard deviation of 100. Who did better? 8

9 9 Example SAT scores for each section are standardized so scores follow an approximately normal distribution with mean 500 points and standard deviation 100 points. We can write this as X~N(500, 100). The maximum score possible is 800 points. a) Sketch the distribution and locate the mean and change of curvature points. b) What proportion of students will score below 700 in one section of the SAT? c) What proportion of students will score below 260 in one section of the SAT? 9

10 10 d) What proportion of students will score between 260 and 700 in one section of the SAT? e) What score corresponds to the bottom 8%? f) Between what two values will you find the central 95% of scores? 10

11 11 g) Between what two values will you find the central 90% of scores? 11

12 12 Section 5.4 Sampling Distribution Review Parameter Numerical summary of the population; this value is usually unknown. Examples:,, 2. Statistic Numerical summary of the sample; this value is usually known. Examples: X, s, 2 s. Sampling Distribution- Shows what values of the statistic occurred and how often they occurred. The Sampling Distribution of a Statistic The probability distribution that specifies the probabilities for all the possible values that the statistic can take. We want to look for a pattern that emerges when we take repeated, random samples and compute a statistic from each sample. The statistics computed from different samples will vary, so they are Random Variables. Random Variables have a distribution; a pattern emerging from repeated sampling. So, when you repeatedly sample and form a statistic for each sample, a pattern will emerge. All sampling distributions will have a mean of the distribution of the statistic and a standard error of the distribution of the statistic. 12

13 13 Concept Consider looking at the last digit in the serial number on a $1 bill. Ie) Let X = the last digit in the serial number on a $1 bill. a) Sketch the Probability Distribution of X. -Uniform -Center = =4.5 -Each dot represents one digit in each dollar bill. b) Suppose that we get the average serial number for the last digit on a one dollar bill for 20 randomly chosen people in our class and observe a sample mean of 4. c) Now suppose that we get the average serial number for the last digit on a one dollar bill for another 20 randomly chosen people in our class and observe a sample mean of 5.2. d) And again suppose that we get the average serial number for the last digit on a one dollar bill for yet another 20 randomly chosen people in our class and observe a sample mean of 6.4. e) Sketch the Sampling Distribution of the sample mean. -Bell Shaped -Centered at 4.5 with a smaller standard deviation. 13

14 14 Sampling Distribution of the Sample Mean X Central Limit Theorem For a random and representative sample (SRS) with large sample size (n 30) or originally normal population, the sampling distribution of the sample mean is approximately Normal with: mean (same as the original distribution) and standard error Notation: X ~ N(, n n ) (original standard deviation divided by the square root of the sample size). Assumptions: Original population is normal or n 30. Z-Score: Z = x n ~ N (0, 1) Comment: The book uses the following notations: = x = x n X ~ N( x x x Z x., ) x 14

15 15 Example The distribution of lawyer salaries at a law firm has a mean of $90,000 and a standard deviation of $50,000. A random sample of 40 lawyers is taken at this firm, and their mean salary is $100,000. a) What is the population mean and standard deviation? b) What is the sampling distribution of the sample statistic? c) What is the Z-Score? 15

16 16 When looking at the Concept example for the sample mean, we saw three types of distributions: 1. Population Distribution probability distribution from which we sample. It is described by parameters ( or p) which are usually unknown. 2. Data Distribution is what we see in practice, when we collect data. We compute sample statistics ( X and p ^ ) to describe the samples and to estimate the population parameters. The larger our sample size, n, the better able we are to do this. 3. Sampling Distribution is the distribution of the sample statistics ( X and ^ p ). 16

17 17 Example In 2000, according to the U.S. Census Bureau, the number of people in a household had a mean of 2.6 and a standard deviation of 1.5. This is based on census information for the population. Suppose the Census Bureau instead had estimated this mean using a random sample of 225 homes. Suppose the sample had a sample mean of 2.4 and standard deviation of 1.4. a) Identify the random variable X. b) Describe the center and spread of the population distribution. c) Describe the center and spread of the data distribution. d) Describe the center and spread (standard error) of the sampling distribution of the sample mean for 225 homes. 17

18 18 Example The distribution of lawyer s salaries at a firm has a mean of $90,000 and standard deviation of $50,000. What is the probability that the average salary of a random sample of five lawyers from this firm is less than $100,000? a) What is the sampling distribution of the statistic? b) Can we answer the question posed? If so, answer the question and if not explain why. 18

19 19 Example The distribution of lawyer s salaries at a firm has a mean of $90,000 and standard deviation of $50,000. What is the probability that the average salary of a random sample of forty lawyers from this firm is less than $100,000? a) What is the sampling distribution of the statistic? b) Can we answer the question posed? If so, answer the question and if not explain why. 19

20 20 Example Machines fill bottles of cola. The bottle label says each one contains 295 ml, but there will be variability in the contents it is a random variable. The distribution of the contents of these bottles is approximately normal with mean 298ml, standard deviation 3 ml. a) Find the probability that one randomly selected bottle contains less than 295 ml. b) Find the probability that the average of a randomly selected six pack is less than 295 ml. 20

21 21 c) Between what two values would you find the average contents of the central 95% of six-packs? d) Why could we use the normal table to find probabilities in this problem? 21

22 22 Section 5.5 The Normal Approximation of the Binomal Distribution Given a binomial setting, we can use the normal distribution to find probabilities instead of the binomial distribution under certain conditions. Let x be a binomial random variable with n trials and probability of success p. Then the probability distribution of x is approximated using a normal curve with = np and np ( 1 p) Notation: X ~ N ( np, np ( 1 p) ) Conditions: We need np > 5 and n(1-p) > 5. This ensures that n is large enough and that p is not too close to zero or one. 22

23 23 Steps to Calculate Probabilities 1. Check assumptions. 2. Calculate = np and np ( 1 p) 3. Convert X=x Z = x.5 np(1 np p) 4. Look for the z-score along the margin of the Table to find the corresponding probability in the middle of the z-table. 23

24 24 Example In a certain population, 15% of the people have Rh-negative blood. A blood bank serving this population receives 92 blood donors on a particular day. a) Using the Binomial Distribution, what is the probability that 10 or fewer are Rh- Negative? b) Using Normal approximation to the Binomial Distribution, what is the probability that 10 or fewer are Rh-Negative? 24

25 25 c) Using the Binomial distribution, what is the probability that 15 to 20 (inclusive) of the donors are RH-negative? d) Using Normal approximation to the Binomial distribution, what is the probability that 15 to 20 (inclusive) of the donors are RH-negative? 25