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Section 6 – 7A: Sampling Distribution of the Sample Means X To Create a Sampling Distribution of the Sample Means X take every possible sample of size n from the distribution of x values and then find the mean of each sample. Each mean is listed as an x a The Distribution of the Sample Means X is the collection of all the sample means x a x 1 + x 2 + ....+ x n = x1 n x1 x2 x3 x4 x5 . . . xa x the population of all x values x 2 + x 5 + ....+ x n = x2 n x1 x 3 + x 4 + ....+ x n = x3 n x3 x 5 + x 7 + ....+ x n = x4 n : : : Take all the possible samples of size n from the distribution of x values and find the mean of each sample the mean of the x values = µ x the SD of the x values = σ x x2 x4 x5 . . . xa X the population of all the possible means of samples of size n taken from the distribution of x values The mean of X is written as ux and ux = ux The standard deviation X is written as σ x and σ σx = x n When is X Normal 1. The distribution of all the possible sample means X is normal if the original population of x values is known to be normal. 2. The distribution of all the possible sample means X is normal if the sample size that created the X distribution is greater than 30. ( n > 30 ) 6 – 7A Central Limit Theorem Lecture Page 1 of 5 © 2012 Eitel Using the Standard Normal Distribution (Z curve) to find Probabilities for a Sampling Distribution of the Means X Assumptions that must be true before you can use a Standard Normal Distribution (Z) to answer questions about a Sampling Distribution of X Original Distribution of x values 1. The original distribution of x values is a collection of data values ( x values) that represents measurements of a population parameter. The distribution of all the x values in the population will be called the “original distribution”. Create a new distribution of Sample Means X 2. A new distribution exits that contains all the possible sample means of size n. A sample of size n is taken from the original population of x values and the mean of that first sample is found ( x 1 ). A second sample of size n is taken from the original population of x values and the mean of that second sample is found ( x 2 ). A third sample of size n is taken from the original population of x values and the mean of that third sample is found ( x 3 ). If we take every possible different sample of size n from the population of x values and compute the mean of each different sample we would have a new population of all the possible sample means. This new population is called the distribution of the sample means X . X Must be Normal 3. The Distribution of Sample Means X must be normal. X will be normal if the original distribution of x values is normal OR the sample size n is greater than 30. n > 30. 6 – 7A Central Limit Theorem Lecture Page 2 of 5 © 2012 Eitel The relationships between the original distribution of x values and the Distribution of Sample Means X The Original distribution of x values must be Normal or n > 30 The Distribution of Sample Means _ µ x = µx X σx σx = n 1. The original distribution of x values will have a mean of µ x and a standard deviation of σ x . Both of these values will be known. µ x is known σ x is known x 2. The mean of the distribution of all possible sample means X has the symbol µ x and is equal to the mean of the original x distribution. µ x = µx 3. The standard deviation of the distribution of all possible sample means X has the symbol σ x and is found by dividing the standard deviation of the original distribution of x values divided by the square root of the sample size: σ σx = n The Relationships between X values and Z values 4. Every x 1 value in the Distribution of Sample Means X has a unique corresponding z value in the Standard Normal Distribution (Z). Convert any x 1 value to its corresponding Z1 value in a Standard Normal Distribution by the formula below. Standard Normal Distribution Z Distribution of Sample Means X Convert x 1 to its unique Z value by z1 = x 1 = ?? µ x = µx σ σx = x n _ X 6 – 7A Central Limit Theorem Lecture (x 1 − µ x ) ⎛σx ⎞ ⎜ ⎟ ⎝ n⎠ Page 3 of 5 z1 = ?? 0 1 Z © 2012 Eitel Relationships between the areas under a Distribution of the Sample Means curve X and the areas under a Standard Normal Z Curve 1. If an x 1 value has a corresponding z 1 value then the area to the left of x 1 is equal to the area to the left of the corresponding z1 value. The area to the left of z1 The area to the left of x1 _ X x1 Z z1 2. If an x 1 value has a corresponding z1 value then the area to the right of x 1 is equal to the area to the right of the corresponding z1 value. The area to the right of The area to the right of z1 x1 _ X x1 z1 Z 3. If two x 1 and x 2 values have corresponding z1 and z2 values then the area between x 1 and x 2 is equal to the area between the corresponding z1 and z2 values. The yellow area between z1 and z2 The area between x 1 and x 2 x1 x2 6 – 7A Central Limit Theorem Lecture _ X z1 Page 4 of 5 z2 Z © 2012 Eitel The Relationship between the area under a Distribution of Sample Means X and Probabilities about selecting n values from a Normal Distribution of x values 1. The area in the left tail of the Distribution of Sample Means X represents the probability of selecting n values from a Normal Distribution of x values and having the mean of those n values x be less than the x 1 value. P( x < x1 ) The yellow area to the left of x 1 represents P( x < x1 ) _ X x1 2. The area in the right tail of the Distribution of Sample Means X represents the probability of selecting n values from a Normal Distribution of x values and having the mean of those n values x be more than the x 1 value. P( x > x1 ) The yellow area to the right of x 1 represents P( x > x1 ) _ X x1 3. The area between two values x 1 and x 2 from a Distribution of Sample Means X represents the probability of selecting n values from a Normal Distribution of x values and having the mean of those n values x be between the two X values x 1 and x 2 . P( x1 < x < x 2 ) The yellow area between x 1 and x 2 represents P( x1 < x < x 2 ) x1 6 – 7A Central Limit Theorem Lecture x2 Page 5 of 5 _ X © 2012 Eitel