Download
slide1 n.
Skip this Video
Loading SlideShow in 5 Seconds..
Sampling Distribution of the Sample Mean PowerPoint Presentation
Download Presentation
Sampling Distribution of the Sample Mean

Sampling Distribution of the Sample Mean

122 Vues Download Presentation
Télécharger la présentation

Sampling Distribution of the Sample Mean

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

  1. Sampling Distribution of the Sample Mean

  2. Example Let X denote the lifetime of a battery Suppose the distribution of battery battery lifetimes has • mean lifetime,  = 400 hours • standard deviation of lifetimes, s = 40 hours

  3. A Sample of Size n is Taken • Calculate the mean of these n batteries • Then another sample of size n batteries is taken • The mean of this second sample of n batteries is calculated • This is done again, and again, and again, and ….

  4. X (Life of a battery) is normal, and • σ is known Distribution of

  5. Even if X does not have a normal distribution, will be approximately normal if n is large. Central Limit Theorem n = 30 is usually large enough to use this approximation.

  6. ExampleWhen Distribution of X is Normal X = the life of abattery • Assume battery life is: • Distributed normal • Mean battery life  = 400 hours • Standard deviation of battery life  = 40 hours • We choose a battery at random • The battery lasts 350 hours • This is an observation of X

  7. X x = 350 Observation of X  = 40 μ = 400

  8. The Random Variable • Suppose random samples of size n = 4 batteries are selected independently and their sample means calculated • These are observations of the random variable

  9. AN OBSERVATION OF • Suppose 4 batteries are selected and their lives are: 420, 450, 380, 390 • Their average = (420 + 450 + 380 + 390)/4 = 410 • This is an observation of a random variable for the Sample Mean

  10. DISTRIBUTION OF WHEN X IS NORMAL

  11. _X Sampling Distribution and anObserved Value

  12.  = 400 x = 350 Observation of XWhen X is Not Normal f(x)  = 40 X

  13. AN OBSERVATION OF • Suppose 4 batteries are selected and their lives are: (410, 450, 360, 360) • Their average = (410 + 450 + 360 + 360)/4 = 395 • This is an observation of a random variable for the Sample Mean • But the sample size is small so we do not know the distribution of -- we can’t plot it

  14. Using A Larger Sample Size • Suppose 100 batteries are selected and their lives are: (420, 450, 380, 350,…., 415) • Their average = (420 + … + 415)/100 = 408 • This is an observation of the random variable (where n = 100) • Because this is a large sample, the distribution of is approximately normal

  15. _X Sampling Distribution and anObserved Value

  16. EXAMPLES • Answer the following questions assuming: • Battery life is distributed normal • Battery life distribution is not normal (or unknown) • What is the probability: • a random battery will last longer than 408 hours? • the average of life of 16 batteries be longer than 408 hours? • the average life of 100 batteries will be longer than 408 hours?

  17. 1 - .5793 = .4207 .5793 408 P(X > 408)X normal  = 40 400 X 0 Z .20

  18. ? P(X >408)X Not Normal  = 40  = 400 x = 408 X

  19. 1 - .7881 = .2119 _X 400 408 _ P(X >408) X Normal, n = 16 .7881 0 Z .80

  20. _ P(X >408) X Not Normal, n = 16 • Can’t do • X is Not Normal and n is small • So we do not know the distribution of the Sample Mean

  21. 1 - .9772 = .0228 _X 400 408 _ P(X >408) X Normal, n = 100 .9772 2.00 0 Z

  22. 1 - .9772 = .0228 _X 400 408 _ P(X >408) X Not Normal, n = 100 .9772 2.00 0 Z

  23. Using Excel • As long as it can be assumed that the distribution of the sample mean is normal, NORMDIST and NORMINV can be used to give probabilities except: • Instead of using , put in /n • Let Excel do the arithmetic • Example: Find the probability the average of 100 batteries exceeds 408 hours

  24. Review • Given a distribution X, with  and  known -- for samples of size n: • If X is normal and  is known • If X is not normal