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Chapter 10 Statistical Inference: One- Sample Hypothesis Test I Scientific Hypothesis

Chapter 10 Statistical Inference: One- Sample Hypothesis Test I Scientific Hypothesis  A testable supposition that is tentatively adopted to account for certain facts and to guide in the investigation of others. . II Statistical Hypotheses

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Chapter 10 Statistical Inference: One- Sample Hypothesis Test I Scientific Hypothesis

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  1. Chapter 10 Statistical Inference: One- Sample Hypothesis Test I Scientific Hypothesis A testable supposition that is tentatively adopted to account for certain facts and to guide in the investigation of others.

  2. II Statistical Hypotheses Statements about one of more parameters of a population distribution A. Null Hypothesis (H0) Hypothesis that the researcher tests B. Alternative Hypothesis (H1) Hypothesis that corresponds to the researcher’s scientific hunch

  3. C. Examples of Null and Alternative Hypotheses 1.  denotes the unknown mean of a population; 0 denotes a hypothesized value of the population mean that is to be tested.

  4. D. Characteristics of the Null Hypothesis, H0, and the Alternative Hypothesis, H1 1. H0 contains an exact statement,  = 0, that can be tested. H1 is an inexact statement:  > 0,  < 0, or  ≠ 0. 2. The null and alternative hypotheses are mutually exclusive and exhaustive. If the null hypothesis is rejected, the alternative is probably true.

  5. E. Statistical Hypotheses for An Experiment 1. The dean at Idle-On-In College believes that a new registration procedure will reduce the time required for students to register. For the past several years, the current procedure has required 3.10 hours. The statistical hypotheses are where denotes the unknown population mean for the new procedure and 0 = 3.10 denotes the mean of the current procedure.

  6. 3. To test the null hypothesis, the dean conducts a trial run with a random sample of n = 27 students and obtains the following data: and 4. The new procedure appears to require less time than the old procedure. But the dean worries because she knows that sample means vary from sample to sample. She wonders, “Is it possible that the population mean of the new procedure is really equal to or greater than the old procedure?”

  7. 4. To determine if the null hypothesis is credible or not credible, the dean computes a t test statistic for her data and determines the probability of obtaining a sample mean of 2.90 if the true population mean is really equal to or greater than 3.10. 5. She discovers that the probability of obtaining a a sample mean of 2.90 if 0 = 3.10 is .001. She decides to reject the null hypothesis and concludes that the new procedure is better.

  8. F. Role of Logic in Hypothesis Testing

  9. III Two Statistics for Testing H0:  = 0 A. t Test Statistic

  10. B. Sampling Distribution of the t Statistic • Figure 1. Graph of the distribution of t for 4, 12, and ∞ • degrees of freedom. When n=∞, the t distribution is • identical to the z (standard normal) distribution.

  11. 1. Concept of degree of freedom, df or   Degrees of freedom for the t statistic:  = n – 1 3. Variance of the t statistic: 4. A bit of history: 1908, the beginning of a new era in statistics 5. Contribution of William Sealey Gossett (Student)

  12. Table 1. Percentage Points of Student’s t Distribution Level of Significance for a One-Tailed Test .05 .025 .01 .005 Degrees of Level of Significance for a Two-Tailed Test Freedom,  .10 .05 .02 .01 10 1.812 2.228 2.764 3.169 19 1.729 2.093 2.539 2.861 20 1.725 2.086 2.528 2.845 26 1.706 2.056 2.479 2.779 30 1.697 2.042 2.457 2.750 38 1.686 2.024 2.429 2.712 ∞ 1.645 1.960 2.326 2.576

  13. C. z Test Statistic 1. The z test statistic is rarely used because the population standard deviation is usually not known.

  14. 2. The sampling distribution of z is the standard normal distribution. 3. Var(z) = 1 D. Comparison of t and z − −

  15. IV Steps in Testing a Null Hypothesis A. Registration Example at Idle-on-in College 1. Is a new registration procedure better than the current procedure? For the past several years, the current procedure has required 3.10 hours. 2. To evaluate the new procedure, the dean conducts a trial run with a random sample of n = 27 students and obtains the following data: and

  16. 3. Is the new procedure, really better than the current procedure, 4. Because of sampling variability, it is unlikely that the sample mean is equal to the population mean of the new procedure. 5. It seems reasonable to reject the null hypothesis if the observed sample mean, is very improbable given that 0 is equal to 3.10.

  17. Figure 2. Illustration of sampling variability of sample means for six random samples from a population

  18. 6. A sample mean that would be observed 5 or fewer times in a hundred replications of an experiment (Probability = 5/100 = .05) is considered very improbable. 7. A sample mean with probability 5/100 = .05 is reasonable grounds for rejecting a null hypothesis. This criterion is called a significance level and denoted by  = .05.

  19. B. Five-Steps in Testing a Null Hypothesis for the Registration Example Step 1. State the statistical hypotheses: Step 2. Specify the test statistic: t statistic because she wants to test  ≥ 3.10,  is unknown, the sample is random, and the population distribution of X is probably approximately normal

  20. Step 3. Specify the sample size: n = 27 and the sampling distribution: t distribution with  = n – 1 = 26, because  is unknown and must be estimated, and the population distribution of X is probably approximately normal Step 4. Specify the significance level:  = .05

  21. Step 5. Obtain a random sample of size n, compute t, and make a decision Decision Rule: Reject the null hypothesis if t falls in the lower 5% of the sampling distribution of t; otherwise, do not reject the null hypothesis. If the null hypothesis is rejected, conclude that the new registration procedure reduces the time to register; if the null hypothesis is not rejected, do not draw this conclusion.

  22. C. Selecting a Level of Significance,  1. Common conventions in specifying  D. Specifying the Critical Region for Rejecting H0 1. The location of the critical region is determined by the alternative hypothesis. 2. The size of the critical region is determined by the level of significance, . 3. The critical region identifies values of that are very unlikely if the null hypothesis is true.

  23. Figure 3. Critical region for rejecting the null hypothesis

  24. V Computation of t Statistic Registration Student Time, Xi 1 2.9 0.00 2 2.7 0.04 3 2.4 0.25 . . . 27 2.5 0.16

  25. Figure 4. in the =  critical region

  26. VI Some Experimental Design Considerations A. John Henry Effect B. Importance of Control Groups VII One- and Two-Tailed Tests A. One-Tailed Test 1. Critical region is in either the upper or lower tail of the sampling distribution.

  27. Figure 5. Critical region for H0: ≤ 3.10

  28. B. Two Tailed Test Figure 6. For H0: = 3.10, the critical region is divided between the upper and lower tails of the t distribution.

  29. C. Denoting Critical Values of t 1. Upper and lower two-tailed critical values are denoted by, respectively, t/2, and–t/2, , where is the significance level and  is the degrees of freedom. 2. t, denotes the upper one-tailed critical value 3. –t, denotes the lower one-tailed critical value

  30. D. Merits of One- and Two-Tailed Tests VIII Type I and Type II Errors A. Type I Error, corresponds to , (Rejecting a True Null Hypothesis) B. Type II Error, corresponds to , (Failing to Reject a False Null Hypothesis)

  31. C. Correct Acceptance and Correct Rejection D. Power, 1 –, for the Registration Example 1. Value of the sample mean that cuts off the lower .05 region of the t sampling distribution

  32. 2. Minimum reduction in registration time that would warrant changing to the new procedure where  = 2.95 corresponds to the new registration time. 3. The size of the region corresponding to a Type II error can be determined by computing a t statistic for the difference

  33. 4. t statistic for computing the probability of a Type II error, , and power, 1 – 5. Area above t = 0.880 is .19 (Prob. of a Type II error)

  34. Table 2. Probabilities Associated with the Decision Process ≤

  35. E. Factors that determine the probability of a Type II error,  and power, 1 –  1. Level of significance,  2. Size of the sample, n 3. Size of the population standard deviation,  4. Magnitude of the difference between m and m.

  36. 5. Simple way to increase power: increase the sample size. IX Determining the Required Sample Size (n) A. Cohen’s Effect Size, d   

  37. Table 3. Approximate n Required for Testing Hypotheses d        0.2 .05 156 215 272 198 264 326 .01 253 328 396 294 374 447 0.5 .05 27 36 45 34 44 54 .01 43 55 66 51 63 75 0.8 .05 12 15 19 15 19 22 .01 19 24 28 22 27 32

  38. B. Statistical Significance Versus Practical Significance C. Reporting p Values 1. Obtaining p values from Excel Select INSERT from the Excel menu Select Function Select the TDIST function

  39. 2. TDIST function TDIST(x,deg_freedom,tails)  Replace x with the t value  Replace deg_freedom with the df for t Replace tails with 1 for one-tail and 2 for two- tail TDIST(3.449,26,1) The p value is .0009652, or < .001

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