x 1 + x x n n = x 1 x 2 + x x n n = x 2 x 3 + x x n n = x 3 x 5 + x x n = x n

Transcription

1 Sectio 6 7A Samplig Distributio of the Sample Meas To Create a Samplig Distributio of the Sample Meas take every possible sample of size from the distributio of x values ad the fid the mea of each sample Each mea is listed as a x a The Distributio of the Sample Meas is the collectio of all the sample meas x a x = + x x = x 3 x 4 x 5 x a x the populatio of all x values the mea of the x values = µ x the SD of the x values = σ x x 3 + x x = x 3 x 5 + x x = x 4 Take all the possible samples of size from the distributio of x values ad fid the mea of each sample x 3 x 4 x 5 x a the populatio of all the possible meas of samples of size take from the distributio of x values The mea of is writte as u x ad u x = u x The stadard deviatio is writte as σ x ad Whe is Normal 1 The distributio of all the possible sample meas is ormal if the origial populatio of x values is kow to be ormal 2 The distributio of all the possible sample meas is ormal if the sample size that created the distributio is greater tha 30 ( > 30 ) 6 7A Cetral Limit Theorem Lecture Page 1 of Eitel

2 Usig the Stadard Normal Distributio ( curve) to fid Probabilities for a Samplig Distributio of the Meas Assumptios that must be true before you ca use a Stadard Normal Distributio () to aswer questios about a Samplig Distributio of Origial Distributio of x values 1 The origial distributio of x values is a collectio of data values ( x values) that represets measuremets of a populatio parameter The distributio of all the x values i the populatio will be called the origial distributio Create a ew distributio of Sample Meas 2 A ew distributio exits that cotais all the possible sample meas of size A sample of size is take from the origial populatio of x values ad the mea of that first sample is foud ( ) A secod sample of size is take from the origial populatio of x values ad the mea of that secod sample is foud ( ) A third sample of size is take from the origial populatio of x values ad the mea of that third sample is foud ( x 3 ) If we take every possible differet sample of size from the populatio of x values ad compute the mea of each differet sample we would have a ew populatio of all the possible sample meas This ew populatio is called the distributio of the sample meas Must be Normal 3 The Distributio of Sample Meas must be ormal will be ormal if the origial distributio of x values is ormal OR the sample size is greater tha 30 > A Cetral Limit Theorem Lecture Page 2 of Eitel

3 The relatioships betwee the origial distributio of x values ad the Distributio of Sample Meas The Origial distributio of x values must be Normal or > 30 The Distributio of Sample Meas µ x is kow σ x is kow x 1 The origial distributio of x values will have a mea of µ x ad a stadard deviatio of σ x Both of these values will be kow 2 The mea of the distributio of all possible sample meas has the symbol µ x ad is equal to the mea of the origial x distributio 3 The stadard deviatio of the distributio of all possible sample meas has the symbol σ x ad is foud by dividig the stadard deviatio of the origial distributio of x values divided by the square root of the sample size σ x = σ The Relatioships betwee values ad values 4 Every value i the Distributio of Sample Meas has a uique correspodig z value i the Stadard Normal Distributio () Covert ay value to its correspodig 1 value i a Stadard Normal Distributio by the formula below Distributio of Sample Meas Stadard Normal Distributio =?? Covert to its uique value by = µ x σ x =?? A Cetral Limit Theorem Lecture Page 3 of Eitel

4 Relatioships betwee the areas uder a Distributio of the Sample Meas curve ad the areas uder a Stadard Normal Curve 1 If a value has a correspodig value the the area to the left of is equal to the area to the left of the correspodig value the left of the left of 2 If a value has a correspodig value the the area to the right of is equal to the area to the right of the correspodig value the right of the right of 3 If two ad values have correspodig ad z 2 values the the area betwee ad is equal to the area betwee the correspodig ad z 2 values The area betwee ad The yellow area betwee ad z 2 z 2 6 7A Cetral Limit Theorem Lecture Page 4 of Eitel

5 The Relatioship betwee the area uder a Distributio of Sample Meas ad Probabilities about selectig values from a Normal Distributio of x values 1 The area i the left tail of the Distributio of Sample Meas represets the probability of selectig values from a Normal Distributio of x values ad havig the mea of those values x be less tha the value P( x < ) The yellow area to the left of represets P x < 2 The area i the right tail of the Distributio of Sample Meas represets the probability of selectig values from a Normal Distributio of x values ad havig the mea of those values x be more tha the value P( x > ) The yellow area to the right of represets P x > 3 The area betwee two values ad from a Distributio of Sample Meas represets the probability of selectig values from a Normal Distributio of x values ad havig the mea of those values x be betwee the two values ad P( < x < ) The yellow area betwee ad represets P( < x < ) 6 7A Cetral Limit Theorem Lecture Page 5 of Eitel