Quantitative Methods for Business COMM5005 Lecture 7 Yiyuan Xie Statistics Flow Chart Statistics Disclaimer:
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prohibited. Descriptive Inferential Multiple linear regression Moments First Moment: Mean Second Moment: Variance Third Moment: Skewness Fourth Moment: Kurtosis Distributions of Variables Normal Distribution Binomial Distribution Bi-variate Distribution Uniform Distribution Lecture 8 Probability Simple linear regression Hypothesis Testing Confidence Interval Estimation Sampling distributions Lecture 7 Lecture 6Week 5 Lecture 6 In this lecture we will cover: Confidence interval estimation Fundamentals of hypothesis testing: One-sample tests 4-2 Lecture 7 topics Objectives Construct and interpret confidence interval estimates Determine the sample size necessary to develop a confidence interval Identify the basic principles of hypothesis testing Explain the assumptions of hypothesis-testing procedure Conduct a hypothesis test Readings 4-4 Sections of Berenson, M. et al. 5th ed. will help you to understand this week s topics more clearly. Chapter Name Pages 8.1 – 8.4 Confidence interval estimation 279-299 9.1-9.4 Fundamentals of hypothesis testing: One-sample tests 315-340 1. Confidence interval estimation Suppose you want to know the mean number of hours of paid work undertaken per week by UNSW students. What you can do is to conduct a survey. The survey shows you a sample mean of 14.8 hours. But the question is, how accurate is the answer of 14.8 hours We need a confidence interval to answer this question. A point estimate is the value of a single sample statistic. A confidence interval provides a range of values constructed around the point estimate. Confidence interval (1 of 3) An interval gives a range of values Takes into consideration variation in sample statistics from sample to sample Based on observations from one sample Gives information about closeness to unknown population parameters Stated in terms of level of confidence Can never be 100% confident Confidence interval (2 of 3) The general formula for all confidence intervals is: Point Estimate ± (Critical Value)*(Standard Error) This represents confidence for which the interval will contain the unknown population parameter. How to determine the critical value It depends on how confident we want to be. A higher confidence level larger critical value. Confidence interval (3 of 3) Common confidence levels = 90%, 95% or 99% Also written (1 – ) = 0.90, 0.95 or 0.99 A relative frequency interpretation: In the long run, 90%, 95% or 99% of all the confidence intervals that can be constructed (in repeated samples) will contain the unknown true parameter A specific interval either will contain or will not contain the true parameter: Example: suppose [12.2, 17.4] is the 95% confidence interval you constructed for the paid work example in the beginning of the slides. You can say “I am 95% confident that the mean work hours in the population of UNSW students is somewhere between 12.2 and 17.4 hours.” Confidence interval for μ (σ known) X nσ/ n σ ZX Finding the critical Z value The value of Z needed for constructing a confidence interval is called the critical value for the distribution. Critical value: The value in a distribution that cuts off the required probability in the tail for a given confidence level. For a 95% confidence interval the value of α is 0.05. The critical Z value corresponding to a cumulative area of 0.9750 is 1.96 because there is 0.025 in the upper tail of the distribution and the cumulative area less than Z = 1.96 is 0.975. There is a different critical value for each level of confidence 1 – α. Normal curve for determining the Z value needed for 95% confidence Normal curve for determining the Z value needed for 99% confidence Common levels of confidence Confidence Level Confidence Coefficient 1-α Z Value 80% 0.80 1.28 90% 0.90 1.645 95% 0.95 1.96 98% 0.98 2.33 99% 0.99 2.576 99.8% 0.998 3.08 99.9% 0.999 3.27 Example Confidence interval for μ (σ unknown) If the population standard deviation σ is unknown, we can substitute the sample standard deviation, S This introduces extra uncertainty, since S is variable from sample to sample So we use the Student’s t distribution instead of the normal distribution the t value depends on degrees of freedom denoted by sample size minus 1; i.e. (d.f = n – 1) d.f are number of observations that are free to vary after sample mean has been calculated Confidence interval for μ (σ unknown) Confidence interval estimate where t is the critical value of the t distribution with n – 1 degrees of freedom and an area of α/2 in each tail Note that S is the standard deviation of the sample: n S tX 1-n 1-n )X(X S n 1i 2 i Degrees of freedom Degrees of freedom: the number of values in the calculation of a statistic that are free to vary. Suppose the mean of 3 numbers is 8.0 Let X1 = 7 Let X2 = 8 What is X3 If the mean of these three values is 8.0, then X3 must be 9 (i.e. X3 is not free to vary) Here, n = 3, so degrees of freedom = n – 1 = 3 – 1 = 2 (2 values can be any numbers, but the third is not free to vary for a given mean) t distribution with 99 degrees of freedom Application: confidence interval estimation for the proportion n )(1 σp Confidence interval endpoints Upper and lower confidence limits for the population proportion are calculated with the formula Z is the standard normal value for the level of confidence desired p is the sample proportion n is the sample size n p)p(1 Zp Example A random sample of 100 people shows that 25 are left-handed The point estimate of the proportion is 25/100. Form a 95% confidence interval for the true proportion of left- handers: /1000.25(0.75)1.9625/100 p)/np(1Zp 0.3349 0.1651 (0.0433) 1.96 0.25 Interpretation We are 95% confident that the true percentage of left-handers in the population is between 0.1651 and 0.3349; i.e. 16.51% and 33.49% Although the interval from 0.1651 to 0.3349 may or may not contain the true proportion, 95% of intervals formed from repeated samples of size 100 in this manner will contain the true proportion. Determining sample size: general guideline n σ ZX Determining sample size for the mean The sample size n is equal to the product of the Z value squared and the variance σ2, divided by the sampling error e squared. To determine the required sample size for the mean, you must know the: desired level of confidence (1 – ), which determines the critical Z value acceptable sampling error, e standard deviation, σ 2 22 e σZ n If σ is unknown If unknown, σ can be estimated using one of the following approaches: from past data using that data s standard deviation if population is normal, range is approximately 6σ so we can estimate σ by dividing the range by 6 conduct a pilot study and estimate σ with the sample standard deviation, 1-n )X(X S n 1i 2 i Application: determining sample size for the proportion 2 2 e )(1Z n ππ Example: determining sample size for the proportion 450.74 (0.03) 0.12)(0.12)(1(1.96) e )(1Z n 2 2 2 2 ππ 2. Fundamentals of hypothesis testing The null hypothesis, H0 How to formulate a null hypothesis, H0 States the belief or assumption in the current situation (status quo) Begin with the assumption that the null hypothesis is true Refers to the status quo Always contains = , ≤ or sign Is always about a population parameter; e.g. μ, not about a sample statistic Example: The average number of TV sets in Australian homes is equal to 3 (H0 : μ = 3 ) The alternative hypothesis, H1 Alternative hypothesis is the opposite of the null hypothesis, and it is generally the claim or hypothesis that the researcher is trying to prove. How to formulate an alternative hypothesis Challenges the status quo Can only contain either the < , > or ≠ sign (not = ) e.g. The average number of TV sets in Australian homes is not equal to 3 (H1 : μ ≠ 3 ) Determining the test statistic and regions of rejection and non-rejection The test statistic is a value derived from sample data that is used to determine whether the null hypothesis should be rejected or not. The sampling distribution of the test statistic is divided into two regions, a region of rejection and a region of non-rejection. To make a decision concerning the null hypothesis, you first determine the critical value of the test statistic. Regions of rejection and non-rejection in hypothesis testing Risks in decision making using hypothesis testing – type 1 error Type I error Reject a true null hypothesis Considered a serious type of error Example: The Australian Government will invest almost $6 million to support research and development of Australian COVID-19 vaccine. Then testing the vaccine, the null hypothesis: the vaccine has no effect, and the alternative hypothesis is: the vaccine has an effect. Type I error in this example: the test rejects the null hypothesis (i.e. showing an effect), but in fact, the vaccine does not have any effect. The probability of Type I error is Called level of significance of the test; i.e. 0.01, 0.05, 0.10 Set by the researcher in advance Risks in decision making using hypothesis testing Type II error Fail to reject a false null hypothesis Example: In the previous example, the null hypothesis: the vaccine has no effect, and the alternative hypothesis is: the vaccine has an effect. Type II error: the test does not reject the null hypothesis (i.e. showing no effect), but in fact, the vaccine does have any effect. The probability of Type II error is β The power of a statistical test, 1 – β, is the probability that you will reject the null hypothesis when it is false and should be rejected. The level of significance, The level of significance, , defines the unlikely values of the sample statistic if the null hypothesis is true That is, it defines rejection region of the sampling distribution Typical values of are 0.01, 0.05 or 0.10 is selected by the researcher at the beginning It provides the critical value(s) of the test The confidence coefficient The confidence coefficient, 1 – , is the probability that you will not reject the null hypothesis, H0, when it is true and should not be rejected. The confidence level of a hypothesis test is: (1 – ) X 100%. Z Test of Hypothesis for the Mean (σ Known) The Z test of hypothesis for the mean is a test about the population mean which uses the standard normal distribution. n σ μX Z Two-tail tests A two-tail test is a hypothesis test where the rejection region is divided into the two tails of the probability distribution. There are two cut-off values (critical values) that define the regions of rejection. Example: The two-tail test can be used to test the null hypothesis that the average number of TV sets in Australian homes is equal to 3 (H0 : μ = 3 ), because the alternative hypothesis is μ ≠ 3 (that is, either greater or less than 3 – two sides). Critical value approach to testing For a two-tail test for the mean, σ known: Convert sample statistic ( ) to the test statistic (Z statistic) Determine the critical Z values for a specified level of significance from a Table E.2 or computer (important: each tail of the distribution must have an area of /2) Decision Rule: If the test statistic falls in the rejection region, reject H0; otherwise do not reject H0 X The six-step method of hypothesis testing 1 State the null hypothesis, H0, and the alternative hypothesis, H1 2 Choose the level of significance, , and the sample size, n 3 Determine the appropriate test statistic and sampling distribution 4 Determine the critical values that divide the rejection and non- rejection regions 5 Collect data and calculate the value of the test statistic 6 Make the statistical decision and state the managerial conclusion if the test statistic falls into the non-rejection region, do not reject the null hypothesis H0; if the test statistic falls into the rejection region, reject the null hypothesis express the managerial conclusion in the context of the real- world problem The p-value approach to hypothesis testing The p-value is the probability of getting a test statistic more extreme than the sample results, given that the null hypothesis, H0, is true. It is also called observed level of significance It is the smallest value of for which H0 can be rejected Decision making: (1) choose a level of significance ; (2) Compute the sample statistic and use it to obtain the p-value from Table E.2 or computer, then If p-value < , reject H0 If p-value , do not reject H0 Finding a p-value for a two-tail test: suppose the Z statistic is 1.5, so p-value = 0.0668 + 0.0668 = 0.1336 > as 0.05, thus do not reject H0. A connection between confidence interval and hypothesis testing Although confidence interval estimation and hypothesis testing are based on the same set of concepts, they are used for different purposes. Confidence intervals: used to estimate parameters Hypothesis testing: used for making decisions about specified values of population parameters One-tail tests Critical value approach to testing For a one-tail test for the mean, σ known: Convert sample statistic ( ) to the test statistic (Z statistic) Determine the critical Z values for a specified level of significance from a Table E.2 or computer (important: be careful on which tail the region of rejection is in, and the tail of the distribution must have an area of ) Decision Rule: If the test statistic falls in the rejection region, reject H0; otherwise do not reject H0 X A one-tail or directional test is a hypothesis where the entire rejection region is contained in one tail of the sampling distribution. The test can be either upper-tail or lower-tail. t test of hypothesis for the mean (σ unknown) The t test of hypothesis for the mean (σ unknown) is a test about the population mean that uses a t distribution. where S is the sample standard deviation, and the test statistic t follows a t distribution having n – 1 degrees of freedom. The procedure of the test is similar to the case where σ is known. The only difference is that you should use a t distribution table (E.3) rather than a normal distribution table (E.2). n S μX t
