Introductory Statistics. Lecture 17

Examples 3 and 4 Math 140 Introductory Statistics Lecture 17 „ Ex3: The Probability of 25 or fewer Children What is the probability that a random sample of 20 families in the United States will have a total of 25 children or fewer? „ Ex4: Reasonably Likely Totals In a random sample of 20 families, what total numbers of children are reasonably likely? Sample Size vs. Population Size 7.3 Sampling Distribution of the Sample Proportion „ As long as the sample size is a small „ You often hear reports of percentages or percentage (around 10% or less) of the population size, it doesn’t matter much if you sample with or without replacement, and, in fact, the population size will have little effect on the statistical analysis. „ (if the sample size is more than 10% of the population size then a more complex formula needs to be used for the standard error, we will not do this here since it rarely happens in practice) proportions: About 60% of automobile drivers in Mississippi use seat belts. (The national average is about 82%.) „ To make intelligent decisions based on data that is reported this way, you must understand the behavior of proportions that arise from random samples. „ The properties of sample proportions are similar to the properties of sample means. 1 number of successes number of successes = n sample size pˆ = 10 5 0 number of successes 10 Center and Spread for Sample Proportions „ The following diagram shows the exact sampling „ We can assign successes as follows: „ Non-user of seatbelt 0 „ User of seat-belt 1 „ We then have the following relative frequency table: Use Seat Belts Relative Frequency In general 1 0.9 0.8 0.7 0.6 0.50 0.40 0.30 0.20 0.10 1 0.00 0.9 0.8 0.7 0.6 0.50 0.40 0.30 0.20 0.10 0.00 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 mean of the population as follows µ = ∑ x ⋅ P (x) = 0(.4) + 1(0.6) „ And in general µ = ∑ x ⋅ P (x ) 0.4 1- p = 0(1 − p ) + 1( p ) = p 1 0.6 p „ On the other hand we know that pˆ = 0.20 „ Then we can calculate the 0 „ And by adding the ones we get 0.10 15 Computer Simulations distributions of the sample proportion for samples of size 10,20, and 40; when p = 0.6 0.00 20 1.00 pˆ = 25 0.80 numbers between 1 and 10. (use your calculator or a random row in the Table D on page 828) „ Count the number of successes the following way: Numbers between 1 and 6 are successes, 7 to 10 (or 0) are not. „ Calculate 0.60 „ Choose 10 random of the population having a certain property. We say that p is the proportion of “successes” according to our property. (e.g. using a seat belt) „ Note that p is always a number between 0 and 1. „ When we select a sample of size n, we calculate the proportion of successes in our sample by dividing the number of successes by the sample size. We call this the sample proportion and we denote it by p-hat. 0.40 „ In a certain population we say that p is the proportion 0.20 Simulation (Activity 7.3a) 0.00 The Sample Proportion p-hat sum of values =x sample size the mean of the sampling distribution of the sample mean is equal to the mean of the population, that is µ pˆ = µx = µ = p 2 Center and Spread for Sample Proportions Use Seat Belts Relative Frequency In general 0 0.4 1- p 1 0.6 p „ Similarly we can calculate the standard deviation of the population as follows σ= ∑ (x − µ ) P (x ) = 2 Center and Spread for Sample Proportions „ On the other hand we know that the standard deviation of the sampling distribution of the sample mean is equal to the SD of the population divided by the square root of the sample size, that is = (0 − 0.6) 2 0.4 + (1 − 0.6) 2 0.6 = (0.6) 2 0.4 + (0.4) 2 0.6 σ pˆ = σ x = = (0.6)(0.4)(0.6 + 0.4) = (0.6)(0.4) and in general σ = ∑ (x − µ ) P (x ) = 2 σ pˆ = σ n = p (1 − p ) n p (1 − p ) n = (0 − p ) 2 (1 − p ) + (1 − p ) 2 p = p (1 − p ) 3