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Probability and Samples • Sampling Distributions • Central Limit Theorem • Standard Error • Probability of Sample Means Inferential Statistics tomorrow and beyond Population Sample last week and today Probability When we take a sample from a population we can talk about the probability of - getting a certain type of individual when we sample once last Thursday - getting a certain type of sample mean when n>1 today Distribution of Individuals in a Population 6 5 4 3 2 1 10 20 30 40 50 60 70 p(X > 50) = ? Distribution of Individuals in a Population 6 5 4 3 2 1 10 20 30 40 50 60 70 p(X > 50) = 1 = 0.11 9 Distribution of Individuals in a Population 6 5 4 3 2 1 10 20 30 40 50 60 70 p(X > 30) = ? Distribution of Individuals in a Population 6 5 4 3 2 1 10 20 30 40 50 60 70 p(X > 30) = 6 = 0.66 9 Distribution of Individuals in a Population 6 normally distributed = 40, = 10 5 4 3 2 1 10 20 30 40 50 60 70 p(40 < X < 60) = ? Distribution of Individuals in a Population 6 5 normally distributed = 40, = 10 4 3 2 1 10 20 30 40 50 60 70 p(40 < X < 60) = p(0 < Z < 2) = 47.7% Distribution of Individuals in a Population 6 normally distributed = 40, = 10 5 4 3 2 1 10 20 30 40 50 60 70 p(X > 60) = ? Distribution of Individuals in a Population 6 5 normally distributed = 40, = 10 4 3 2 1 10 20 30 40 50 60 70 p(X > 60) = p(Z > 2) = 2.3% For the preceding calculations to be accurate, it is necessary that the sampling process be random. A random sample must satisfy two requirements: 1. Each individual in the population has an equal chance of being selected. 2. If more than one individual is to be selected, there must be constant probability for each and every selection (i.e. sampling with replacement). Distribution of Sample Means A distribution of sample means is: the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population. Population 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 Distribution of Sample Means from Samples of Size n = 2 Sample # Scores Mean ( X ) 1 2, 2 2 2 2,4 3 3 2,6 4 4 2,8 5 5 4,2 3 6 4,4 4 7 4,6 5 8 4,8 6 9 6,2 4 10 6,4 5 11 6,6 6 12 6,8 7 13 8,2 5 14 8,4 6 15 8.6 7 16 8.8 8 Distribution of Sample Means from Samples of Size n = 2 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 sample mean We can use the distribution of sample means to answer probability questions about sample means Distribution of Sample Means from Samples of Size n = 2 6 5 4 3 2 1 1 2 3 4 5 6 7 sample mean p( X > 7) = ? 8 9 Distribution of Sample Means from Samples of Size n = 2 6 5 4 3 2 1 1 2 3 4 5 6 7 8 sample mean p( X > 7) = 1 =6% 16 9 Distribution of Individuals in Population 6 5 4 3 2 1 = 5, = 2.24 Distribution of Sample Means 6 5 1 2 3 4 5 6 7 8 9 X = 5, X = 1.58 4 3 2 1 1 2 3 4 5 6 7 8 9 sample mean Distribution of Individuals 6 5 4 3 2 1 = 5, = 2.24 Distribution of Sample Means 6 5 1 2 3 4 5 6 7 8 9 p(X > 7) = 25% X = 5, X = 1.58 4 3 2 1 1 2 3 4 5 6 7 8 9 sample mean p(X> 7) = 6% , for n=2 A key distinction Population Distribution – distribution of all individual scores in the population Sample Distribution – distribution of all the scores in your sample Sampling Distribution – distribution of all the possible sample means when taking samples of size n from the population. Also called “the distribution of sample means”. Distribution of Individuals in Population 6 5 4 3 2 1 = 5, = 2.24 Distribution of Sample Means 6 5 1 2 3 4 5 6 7 8 9 X = 5, X = 1.58 4 3 2 1 1 2 3 4 5 6 7 8 9 sample mean Distribution of Sample Means 6 5 Things to Notice 4 3 2 1. The sample means tend to pile up around the population mean. 2. The distribution of sample means is approximately normal in shape, even though the population distribution was not. 3. The distribution of sample means has less variability than does the population distribution. 1 1 2 3 4 5 6 7 8 9 sample mean What if we took a larger sample? Distribution of Sample Means from Samples of Size n = 3 24 22 20 X = 5, X = 1.29 18 16 14 12 p( X > 7) = 10 8 6 4 2 1 2 3 4 5 6 7 sample mean 8 9 1 =2% 64 Distribution of Sample Means As the sample gets bigger, the sampling distribution… 1. stays centered at the population mean. 2. becomes less variable. 3. becomes more normal. Central Limit Theorem For any population with mean and standard deviation , the distribution of sample means for sample size n … 1. will have a mean of 2. will have a standard deviation of n 3. will approach a normal distribution as n approaches infinity Notation the mean of the sampling distribution X the standard deviation of sampling distribution (“standard error of the mean”) X n Standard Error The “standard error” of the mean is: The standard deviation of the distribution of sample means. The standard error measures the standard amount of difference between x-bar and that is reasonable to expect simply by chance. SE = n Standard Error The Law of Large Numbers states: The larger the sample size, the smaller the standard error. This makes sense from the formula for standard error … Distribution of Individuals in Population 6 5 4 3 2 1 = 5, = 2.24 Distribution of Sample Means 6 5 1 2 3 4 5 6 7 8 9 X = 5, X = 1.58 4 3 X 2 1 1 2 3 4 5 6 7 8 9 sample mean 2.24 1.58 2 Sampling Distribution (n = 3) 24 22 20 X = 5 X = 1.29 18 16 14 12 X 10 8 6 4 2 1 2 3 4 5 6 7 sample mean 8 9 2.24 1.29 3 Clarifying Formulas Population Sample Distribution of Sample Means X X X X N ss N n X ss s n 1 notice 2 X 2 n n Central Limit Theorem For any population with mean and standard deviation , the distribution of sample means for sample size n … 1. will have a mean of 2. will have a standard deviation of n 3. will approach a normal distribution as n approaches infinity What does this mean in practice? Practical Rules Commonly Used: 1. For samples of size n larger than 30, the distribution of the sample means can be approximated reasonably well by a normal distribution. The approximation gets better as the sample size n becomes larger. 2. If the original population is itself normally distributed, then the sample means will be normally distributed for any sample size. normal population non-normal population small n large n X is normal X is normal X is nonnormal X is normal Probability and the Distribution of Sample Means The primary use of the distribution of sample means is to find the probability associated with any specific sample. Probability and the Distribution of Sample Means Example: Given the population of women has normally distributed weights with a mean of 143 lbs and a standard deviation of 29 lbs, 1. if one woman is randomly selected, find the probability that her weight is greater than 150 lbs. 2. if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lbs. Given the population of women has normally distributed weights with a mean of 143 lbs and a standard deviation of 29 lbs, 1. if one woman is randomly selected, find the probability that her weight is greater than 150 lbs. Population distribution z = 150-143 = 0.24 29 0.4052 = 143 150 = 29 0 0.24 Given the population of women has normally distributed weights with a mean of 143 lbs and a standard deviation of 29 lbs, 2. if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lbs. X 29 36 Sampling distribution z = 150-143 = 1.45 4.33 0.0735 = 143 150 = 4.33 0 1.45 Probability and the Distribution of Sample Means Example: Given the population of women has normally distributed weights with a mean of 143 lbs and a standard deviation of 29 lbs, 1. if one woman is randomly selected, find the probability that her weight is greater than 150 lbs. P( X 150) .41 2. if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lbs. P( X 150) .07 Practice Example: Given a population of 400 automobile models, with a mean horsepower = 105 HP, and a standard deviation = 40 HP, 1. What is the standard error of the sample mean for a sample of size 1? 40 2. What is the standard error of the sample mean for a sample of size 4? 20 3. What is the standard error of the sample mean for a sample of size 25? 8 Practice Example: Given a population of 400 automobile models, with a mean horsepower = 105 HP, and a standard deviation = 40 HP, 1. if one model is randomly selected from the population, find the probability that its horsepower is greater than 120. .35 2. If 4 models are randomly selected from the population, find the probability that their mean horsepower is greater than 120 .23 3. If 25 models are randomly selected from the population, find the probability that their mean horsepower is greater than 120 .03

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