AGENDA Sampling distributions Errors Sample sizes Confidence intervals Hypotheses tests Type one and type two errors Hypotheses tests for one mean Hypotheses tests for two means Confidence intervals concerning two means 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 t DISTRIBUTION t distribution is similar in shape to the standard normal distribution Symetric around 0 However, the shape varies according to its degrees of freedom Degrees of Freedom = number in sample - 1 Table 4 – page 587 Right tail probability Degrees of freedom Note that the t distribution converges to the standard normal as n approaches infinity 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 F DISTRIBUTION Distribution that represents the ratio of the sample variances of two normally distributed populations having the same variance Parameters that affect the shape of the distribution Numerator degrees of freedom = n1-1 Denominator degrees of freedom = n2-1 Table 6, page 221. Limited number of right hand side values for F Table 6a is 0.05 Table 6b is 0.01 MAXIMUM ERROR OF ESTIMATE For a given sized random sample what will be the maximum size of the estimate error DETERMINATION OF SAMPLE SIZE For a given level of precision we can also determine how many observations need to be in the sample Rearrage the maximum error of estimate equation Must know alpha, E, and sigma May have to estimate sigma INTERVAL ESTIMATION FOR POP. MEAN WITH SIGMA KNOWN 1-alpha level of certainty that the interval will contain the population mean Alpha is typically 0.01 (99 percent certain) 0.05 (95 percent certain) INTERVAL ESTIMATION FOR POP. MEAN WITH SIGMA UNKNOWN 1-alpha level of certainty that the interval will contain the population mean For “large” (>=30)sample sizes approximate sigma with s Alpha is typically 0.01 (99 percent certain) 0.05 (95 percent certain) INTERVAL ESTIMATION FOR POP. MEAN WITH SIGMA UNKNOWN 1-alpha level of certainty that the interval will contain the population mean For “small” (n<30) sample sizes Alpha is typically 0.01 (99 percent certain) 0.05 (95 percent certain) GENERAL HYPOTHESES PROCEDURE Identify Ho and Ha Determine level of significance (generally 0.05) Determine “critical value” criterion from level of significance Calculate “test statistic” Make decision Fail to reject Ho Reject Ho HYPOTHESES TESTS TWO MEANS Same 5 step hypotheses procedure Large sample Small sample LARGE SAMPLE Variance known Variance unknown TEST FOR DIFFERENCE BETWEEN TWO MEANS VARIANCE KNOWN TEST FOR DIFFERENCE BETWEEN TWO MEANS WHEN VARIANCE UNKNOWN, BUT N>=30 Sample variance can be used to approximate the population variance when n1 and n2 are large DIFFERENT VERSIONS OF THE T-TEST FOR SMALL SAMPLES Two sample t-test assuming equal variances Two sample t-test assuming unequal variances AKA Smith-Satterthwaite Test We will learn how to determine if the variance is equal or unequal in chapter 8 Paired t-test TWO SAMPLE T-TEST EQUAL VARIANCES n1 or n2 or both <30 TWO SAMPLE T-TEST UNEQUAL VARIANCES n1 or n2 or both <30 Unequal variances are accounted for by adjusting the degrees of freedom for the critical value PAIRED T-TEST Natural pairing Before and after Difference random variable CONFIDENCE INTERVALS CONCERNING TWO MEANS Types of confidence intervals Small sample confidence interval Large sample confidence interval Paired sample confidence interval SMALL SAMPLE CONFIDENCE INTERVAL When n1 or n2 or both are <=30 Utilizes the t value for alpha / 2 n1+n2-2 degrees of freedom Sample means and variances LARGE SAMPLE CONFIDENCE INTERVAL When n1 and n2 are >=30 Utilizes the Z value for alpha / 2 Sample means and variances PAIRED SAMPLE CONFIDENCE INTERVAL Requires paired type data Utilizes the t value for alpha / 2 n-1 degrees of freedom Mean and variance of difference