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#21




Hypothesis Testing
Hypothesis Testing
________________________________________ Definitions Null Hypothesis ( H0 ) Statement of zero or no change. If the original claim includes equality (<=, =, or >=), it is the null hypothesis. If the original claim does not include equality (<, not equal, >) then the null hypothesis is the complement of the original claim. The null hypothesis always includes the equal sign. The decision is based on the null hypothesis. Alternative Hypothesis ( H1 or Ha ) Statement which is true if the null hypothesis is false. The type of test (left, right, or twotail) is based on the alternative hypothesis. Type I error Rejecting the null hypothesis when it is true (saying false when true). Usually the more serious error. Type II error Failing to reject the null hypothesis when it is false (saying true when false). alpha Probability of committing a Type I error. beta Probability of committing a Type II error. Test statistic Sample statistic used to decide whether to reject or fail to reject the null hypothesis. Critical region Set of all values which would cause us to reject H0 Critical value(s) The value(s) which separate the critical region from the noncritical region. The critical values are determined independently of the sample statistics. Significance level ( alpha ) The probability of rejecting the null hypothesis when it is true. alpha = 0.05 and alpha = 0.01 are common. If no level of significance is given, use alpha = 0.05. The level of significance is the complement of the level of confidence in estimation. Decision A statement based upon the null hypothesis. It is either "reject the null hypothesis" or "fail to reject the null hypothesis". We will never accept the null hypothesis. Conclusion A statement which indicates the level of evidence (sufficient or insufficient), at what level of significance, and whether the original claim is rejected (null) or supported (alternative). Stats: Hypothesis Testing ________________________________________ Introduction Be sure to read through the definitions for this section before trying to make sense out of the following. The first thing to do when given a claim is to write the claim mathematically (if possible), and decide whether the given claim is the null or alternative hypothesis. If the given claim contains equality, or a statement of no change from the given or accepted condition, then it is the null hypothesis, otherwise, if it represents change, it is the alternative hypothesis. The following example is not a mathematical example, but may help introduce the concept. Example "He's dead, Jim," said Dr. McCoy to Captain Kirk. Mr. Spock, as the science officer, is put in charge of statistically determining the correctness of Bones' statement and deciding the fate of the crew member (to vaporize or try to revive) His first step is to arrive at the hypothesis to be tested. Does the statement represent a change in previous condition? • Yes, there is change, thus it is the alternative hypothesis, H1 • No, there is no change, therefore is the null hypothesis, H0 The correct answer is that there is change. Dead represents a change from the accepted state of alive. The null hypothesis always represents no change. Therefore, the hypotheses are: • H0 : Patient is alive. • H1 : Patient is not alive (dead). States of nature are something that you, as a statistician have no control over. Either it is, or it isn't. This represents the true nature of things. Possible states of nature (Based on H0) • Patient is alive (H0 true  H1 false ) • Patient is dead (H0 false  H1 true) Decisions are something that you have control over. You may make a correct decision or an incorrect decision. It depends on the state of nature as to whether your decision is correct or in error. Possible decisions (Based on H0 ) / conclusions (Based on claim ) • Reject H0 / "Sufficient evidence to say patient is dead" • Fail to Reject H0 / "Insufficient evidence to say patient is dead" There are four possibilities that can occur based on the two possible states of nature and the two decisions which we can make. Statisticians will never accept the null hypothesis, we will fail to reject. In other words, we'll say that it isn't, or that we don't have enough evidence to say that it isn't, but we'll never say that it is, because someone else might come along with another sample which shows that it isn't and we don't want to be wrong. Statistically (double) speaking ... State of Nature Decision H0 True H0 False Reject H0 Patient is alive, Sufficient evidence of death Patient is dead, Sufficient evidence of death Fail to reject H0 Patient is alive, Insufficient evidence of death Patient is dead, Insufficient evidence of death In English ... State of Nature Decision H0 True H0 False Reject H0 Vaporize a live person Vaporize a dead person Fail to reject H0 Try to revive a live person Try to revive a dead person Were you right ? ... State of Nature Decision H0 True H0 False Reject H0 Type I Error alpha Correct Assessment Fail to reject H0 Correct Assessment Type II Error beta Which of the two errors is more serious? Type I or Type II ? Since Type I is the more serious error (usually), that is the one we concentrate on. We usually pick alpha to be very small (0.05, 0.01). Note: alpha is not a Type I error. Alpha is the probability of committing a Type I error. Likewise beta is the probability of committing a Type II error. Conclusions Conclusions are sentence answers which include whether there is enough evidence or not (based on the decision), the level of significance, and whether the original claim is supported or rejected. Conclusions are based on the original claim, which may be the null or alternative hypotheses. The decisions are always based on the null hypothesis Original Claim Decision H0 "REJECT" H1 "SUPPORT" Reject H0 "SUFFICIENT" There is sufficient evidence at the alpha level of significance to reject the claim that (insert original claim here) There is sufficient evidence at the alpha level of significance to support the claim that (insert original claim here) Fail to reject H0 "INSUFFICIENT" There is insufficient evidence at the alpha level of significance to reject the claim that (insert original claim here) There is insufficient evidence at the alpha level of significance to support the claim that (insert original claim here) 
The Following User Says Thank You to Faraz_1984 For This Useful Post:  
Bilal Salim (Wednesday, February 02, 2011) 
#22




Stats: Type of Tests
Stats: Type of Tests
________________________________________ This document will explain how to determine if the test is a left tail, right tail, or twotail test. The type of test is determined by the Alternative Hypothesis ( H1 ) Left Tailed Test H1: parameter < value Notice the inequality points to the left Decision Rule: Reject H0 if t.s. < c.v. Right Tailed Test H1: parameter > value Notice the inequality points to the right Decision Rule: Reject H0 if t.s. > c.v. Two Tailed Test H1: parameter not equal value Another way to write not equal is < or > Notice the inequality points to both sides Decision Rule: Reject H0 if t.s. < c.v. (left) or t.s. > c.v. (right) ________________________________________ The decision rule can be summarized as follows: Reject H0 if the test statistic falls in the critical region (Reject H0 if the test statistic is more extreme than the critical value) Confidence Intervals as Tests ________________________________________ Using the confidence interval to perform a hypothesis test only works with a twotailed test. • If the hypothesized value of the parameter lies within the confidence interval with a 1alpha level of confidence, then the decision at an alpha level of significance is to fail to reject the null hypothesis. • If the hypothesized value of the parameter lies outside the confidence interval with a 1alpha level of confidence, then the decision at an alpha level of significance is to reject the null hypothesis. Sounds simple enough, right? It is. However, it has a couple of problems. • It only works with twotail hypothesis tests. • It requires that you compute the confidence interval first. This involves taking a zscore or tscore and converting it into an xscore, which is more difficult than standardizing an xscore. Hypothesis Testing Steps ________________________________________ Here are the steps to performing hypothesis testing 1. Write the original claim and identify whether it is the null hypothesis or the alternative hypothesis. 2. Write the null and alternative hypothesis. Use the alternative hypothesis to identify the type of test. 3. Write down all information from the problem. 4. Find the critical value using the tables 5. Compute the test statistic 6. Make a decision to reject or fail to reject the null hypothesis. A picture showing the critical value and test statistic may be useful. 6. Write the conclusion. Testing a Single Mean ________________________________________ You are testing mu, you are not testing x bar. If you knew the value of mu, then there would be nothing to test. All hypothesis testing is done under the assumption the null hypothesis is true! I can't emphasize this enough. The value for all population parameters in the test statistics come from the null hypothesis. This is true not only for means, but all of the testing we're going to be doing. Population Standard Deviation Known If the population standard deviation, sigma, is known, then the population mean has a normal distribution, and you will be using the zscore formula for sample means. The test statistic is the standard formula you've seen before. The critical value is obtained from the normal table, or the bottom line from the ttable. Population Standard Deviation Unknown If the population standard deviation, sigma, is unknown, then the population mean has a student's t distribution, and you will be using the tscore formula for sample means. The test statistic is very similar to that for the zscore, except that sigma has been replaced by s and z has been replaced by t. The critical value is obtained from the ttable. The degrees of freedom for this test is n1. If you're performing a ttest where you found the statistics on the calculator (as opposed to being given them in the problem), then use the VARS key to pull up the statistics in the calculation of the test statistic. This will save you data entry and avoid round off errors. General Pattern Notice the general pattern of these test statistics is (observed  expected) / standard deviation. Testing a Single Proportion ________________________________________ You are testing p, you are not testing p hat. If you knew the value of p, then there would be nothing to test. All hypothesis testing is done under the assumption the null hypothesis is true! I can't emphasize this enough. The value for all population parameters in the test statistics come from the null hypothesis. This is true not only for proportions, but all of the testing we're going to be doing. The population proportion has an approximately normal distribution if np and nq are both at least 5. Remember that we are approximating the binomial using the normal, and that the p we're talking about is the probability of success on a single trial. The test statistic is shown in the box to the right. The critical value is found from the normal table, or from the bottom row of the ttable. The steps involved in the hypothesis testing remain the same. The only thing that changes is the formula for calculating the test statistic and perhaps the distribution which is used. General Pattern Notice the general pattern of these test statistics is (observed  expected) / standard deviation. 
The Following 2 Users Say Thank You to Faraz_1984 For This Useful Post:  
Altamash Qureshi (Friday, January 09, 2009), Bilal Salim (Wednesday, February 02, 2011) 
#23




I just wanna add a last but not least general point that in CSS they expect that contestant gone through & well versed in theoretical & conceptual ground.
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A contented mind is the greatest blessing a man can enjoy in this world 
#24




for help in subject combination
plz help in subject combination statistics is my major subject what other subject i have to choose along with statistics in B.Sc my subject is math,stats and physics

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