Mid-Term Review Final Review Statistical for Business (1)(2)

Mid-Term Review Remarks This is a close-book examination. Calculation can be carried out in a calculator. Other computational tools, such as mobile phone, are prohibited. There are 8 questions for a total of 100 points. A formula bank is attached (from Chap. 7). Time: 7 Jun 2010 (Mon) 9:30am-11:30pm

Mid-Term Review Chapter 4 A random variable is a variable that assumes numerical values that are determined by the outcome of an experiment A random variable is a variable that assumes numerical values that are determined by the outcome of an experiment. The probability distribution of a discrete random variable is a table, graph, or formula that gives the probability associated with each possible value that the variable can assume Properties 1.For any value x of the random variable, p(x)  0 2.The probabilities of all the events in the sample space must sum to 1, that is,

Mid-Term Review The Binomial Distribution The Binomial Experiment: 1.Experiment consists of n identical trials 2.Each trial results in either “success” or “failure” 3.Probability of success, p, is constant from trial to trial 4.Trials are independent If x is the total number of successes in n trials of a binomial experiment, then x is a binomial random variable Number of ways to get x successes and (n–x) failures in n trials The chance of getting x successes and (n–x) failures in a particular arrangement

Mid-Term Review If x is a binomial random variable with parameters n and p (so q = 1 – p), then

Mid-Term Review The Poisson Distribution If x = the number of occurrences in a specified interval, then x is a Poisson random variable where x can take any of the values x = 0, 1, 2, 3, … and e = … (e is the base of the natural logs)

Mid-Term Review Chapter 5 Recall: A continuous random variable may assume any numerical value in one or more intervals Use a continuous probability distribution to assign probabilities to intervals of values Properties of f(x): f(x) is a continuous function such that 1.f(x)  0 for all x 2.The total area under the curve of f(x) is equal to 1 Essential point: An area under a continuous probability distribution is a probability

Mid-Term Review The Uniform Distribution The mean  X and standard deviation  X of a uniform random variable x are

Mid-Term Review The Normal Distribution If x is normally distributed with mean  and standard deviation , then the random variable z is normally distributed with mean 0 and standard deviation 1; this normal is called the standard normal distribution.

Mid-Term Review Chapter 6: Distribution of Sample Mean (Central Limit Theorem) Sample mean is approximately normally distributed if:  Sample size is larger than 30 (without assuming the population also has a normal distribution);  Sample size is less than 30, and the population also has a normal distribution. – Sample mean μ Population mean – Mean of sample meanσ– Population standard deviation – Standard deviation of sample mean n- sample size When sample size is large: ~

Mid-Term Review Chapter 7 z-Based Confidence Intervals for a Mean with  Known If a population has standard deviation  (known), and if the population is normal or if sample size is large (n  30), then … … a  ) 100% confidence interval for  is

Mid-Term Review t-Based Confidence Intervals for a Mean:  Unknown If the sampled population is normally distributed with mean , then a  )100% confidence interval for  is Sample standard deviation t  /2 is the t point giving a right-hand tail area of  /2 under the t curve having n – 1 degrees of freedom

Mid-Term Review z-Based Confidence Intervals for a Population Proportion If the sample size n is large*, then a  )100% confidence interval for p is * Here n should be considered large if both

Mid-Term Review Sample Size Determination Sample Size Determination Letting E denote the desired margin of error, so that is within E units of , with 100(1-  )% confidence. If is unknown and is estimated from s If σ is unknown and is estimated from s so that is within E units of , with 100(1-  )% confidence. The number of degrees of freedom for the t  /2 point is the size of the preliminary sample minus 1 If is known If σ is known

Mid-Term Review A sample size will yield an estimate, precisely within E units of p, with 100(1-  )% confidence Note that the formula requires a preliminary estimate of p. The conservative value of p = 0.5 is generally used when there is no prior information on p Sample Size Determination Sample Size Determination for p

Chapter 8: Six Step Model for Hypothesis Tests Step 1: State the null and alternate hypothesis Step 2: Select a level of significance Step 3: Identify a test statistics Step 4: Determine the critical value and rejection region Step 5: Take a sample and compute the value of the test statistics Step 6: Make the statistical decision: Do not reject nullReject null and accept alternate

Mid-Term Review Hypotheses H 0 :   H 0 :   H 1 :   H 1 :   Test Statistic  Known  known Test Statistic  Known  known Rejection Rule Reject H 0 if z > z  Reject H 0 if z z  Reject H 0 if z < - z  One-Tailed Tests about a Population Mean:  0 0 z α =1.65 Reject H 0 Do Not Reject H 0 (Critical value) z 0.05 Level of Significance:α A right-Tailed Test H 1 : μ > μ 0  z α =-1.65 Reject H 0 Do Not Reject H 0 (Critical value) z A left-Tailed Test H 1 : μ < μ 0

Mid-Term Review Hypotheses: H 0 :    H 1 :   Test Statistic  Known Rejection RuleReject H 0 if |z| > z  Two-Tailed Tests about a Population Mean  Reject H 0 Do Not Reject H 0 z z Reject H  0.05 Level of Significance:α

Mid-Term Review Tests about a Population Mean: σ Unknown  Test Statistic:  Test Statistic: σ Unknown This test statistic has a t distribution with n - 1 degrees of freedom.  Rejection Rule H 0 :   Reject H 0 if t > t  H 0 :   Reject H 0 if t > t  H 0 :   Reject H 0 if t < - t  H 0 :   Reject H 0 if t < - t  H 0 :   Reject H 0 if | t | > t  H 0 :   Reject H 0 if | t | > t 

Mid-Term Review Test about a Population Proportion  Hypotheses (where p 0 is the hypothesized value of the population proportion). H 0 : p > p 0 H 0 : p < p 0 H 0 : p = p 0 H 1 : p p 0 H 1 : p ≠ p 0 Left-tailed Right-tailed Two-tailed  Test Statistic Where:  Rejection Rule H 0 : p  p  Reject H 0 if z > z  H 0 : p  p  Reject H 0 if z > z  H 0 : p  p  Reject H 0 if z < -z  H 0 : p  p  Reject H 0 if z < -z  H 0 : p  p  Reject H 0 if |z| > z  H 0 : p  p  Reject H 0 if |z| > z 

Mid-Term Review Chapter 9: Two-Sample Tests Two-Sample Tests Population Means, Independent Samples Means, Related Samples Population 1 vs. independent Population 2 Same population before vs. after treatment Examples:

Mid-Term Review Population means, independent samples σ 1 and σ 2 known The test statistic for μ 1 – μ 2 is: σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal Where t has (n 1 + n 2 – 2) d.f.,

Mid-Term Review Test of Related Populations A paired t test: The test statistic for d is a t statistic, with n-1 d.f.: Tests Means of 2 Related Populations  Paired or matched samples  Repeated measures (before/after) Use difference between paired values d i = X 1i - X 2i ( The ith paired difference is d i ) Where S d is: