AGENDA Populations and sample Sampling distribution of the mean (sigma known) Central Limit Theorem Sampling distribution of the mean (sigma unknown) Sampling distribution of the variance POPULATIONS AND SAMPLE Populations can be either Infinite Finite Samples are usually necessary Impossible to observe infinite population Impractical or uneconomical to observe finite population Random sample Representative of the population Unbiased Tables of random numbers, etc. Sampling distributions are what you get when you observe the random sample means and variances SAMPLING DISTRIBUTION OF THE MEAN (SIGMA KNOWN) Population Mean = u Variance = sigma^2 Sample of the population Mean of sample = x bar X bar is a random variable Every sample will be slightly different Mean of x bar = u Variance of x bar = (sigma^2) / n STANDARD ERROR OF THE MEAN The reliability of the sample mean is measured by the standard error of the mean STANDARDIZED SAMPLE MEAN CENTRAL LIMIT THEORM Regardless of the underlying population, the mean of a sample becomes normally distributed as n approaches infinity Standardized sample mean is standard normally distributed In reality the CLT begins at 25 to 30 EXAMPLE Can of paint covers an average of 513.3 square feet with a std of 31.5 square feet What is the probability that a random sample of 40 cans of paint will paint between 510.0 to 520.0 square feet? EXAMPLE Use the standard normal table to determine the probabilities of obtaining the Z values 1.34 ? 0.9099 -0.66 (note error in book) ?0.2546 0.9099-0.2546=0.6553 SAMPLING DISTRIBUTION OF THE MEAN (SIGMA UNKNOWN) Population Mean = u Variance is unknown Replace sigma with sample standard deviation Sample of the population Mean of the sample is x bar Variance of the sample is (s^2) / n t DISTRIBUTION t distribution is similar in shape to the standard normal distibution Symetric around 0 However, the shape varies according to its degrees of freedom Table 4 – page 587 Right tail probability Degrees of freedom Note that the t distribution converges to the standard normal as n approaches infinity t DISTRIBUTION t DISTRIBUTION for a particular number of degrees of freedom EXAMPLE If overloaded by 20% fuses will blow in an average of 12.40 minutes Random sample of 20 fuses are subjected to a 20% overload Mean time = 10.63 Std=2.48 Support or refute manufactures claims EXAMPLE Using table 4 For 19 degrees of freedom, probability of getting a value of -2.861 is 0.005 t = -3.19 has a probability of less than 0.005 Fuses blow in less than 12.40 minutes SAMPLING DISTRIBUTION OF THE VARIANCE Distribution of sample variance Cannot be less than 0 Shape depends on degrees of freedom v = n – 1 Table 5 Right tail probability Degrees of freedom CHI-SQUARE DISTRIBUTION CHI-SQUARE for a particular number of degrees of freedom EXAMPLE Variance of refractive index is known to be 1.26E-4 Shipment is rejected if the variance of a random sample of 20 is greater than 2.0E-4. Probability of rejecting a sample For 19 df, 0.05 critical value =30.144 30.2 is less than 0.05 probability