FUNDAMENTALS OF INFERENTIAL STATISTICSinferential statistics – drawing conclusions on the population based on the samples. parametric statistics examples: o z-test o t-test o F-test in the analysis of variance (ANOVA) non-parametric statistics examples: o chi-square test Recall that: population – the target data sample – a subset of population group size: A group is considered large if n 30. Otherwise, small. Z – tests are usually used for large groups and t-test for small groups. *z-tests and t-tests, oftentimes, can perform each other’s role in inferential statistics. It only differs in setting the critical value. Remember the Central Limit Theorem. Two types of tests according to direction Directional Test (or “one-tailed test”) - determines if the sample mean is significantly less or greater than the population mean. - determines if a group is better or worse than the other. Non-directional Test (or “two-tailed test”) - determines if the sample mean differs from the population mean. - determines if two groups differ. *The type of test is determined by analyzing the problem, specifically on the last question. Or is depending on the hypothesis. Hypothesis – assumption Null hypothesis (H0) - expresses a non-significant difference between the variable / groups - carries the equal sign: = for two-tailed tests, or for one-tailed tests. Alternative hypothesis (Ha) - expresses a significant difference between the variable / groups. - carries the inequality sign: for two-tailed tests, and < or > for one-tailed tests. Level of significance ( ) - the probability of rejecting the null hypothesis. - a pre-determined value which sets the area of rejection where the conclusion will be based. - usually values are = 0.10, = 0.05, or = 0.01 - = 1 – confidence interval. Table of critical values for t-test Table of critical values for z-test: *Based on the areas under the normal curve. Two-tailed test .01 .05 .10 2.575 1.96 1.645 One-tailed test 2.33 or –2.33 1.645 or – 1.645 1.28 or – 1.28 STEPS IN HYPOTHESIS TESTING: 1. Analyze the problem to determine the null and alternative hypotheses. 2. Determine the test statistic to be used (z-test, t-test, etc) 3. Compute for the statistic value. 4. Find the critical value. 5. If the obtained statistic value is beyond the critical value, “reject H0”, and thus, accepting Ha. Otherwise, “Fail to reject H0”, thus ignoring Ha. a) Does mean age of onset of a certain acute disease for school children differ from 11. If the population standard deviation is 19 lbs. Conclusion: ______________________________________________________________ The mean weight of the sample of 40 persons from the Honolulu Heart Study is 64 kg.5? One-tailed test Two-tailed test Ho: _________________________________________________________________ Ha: _________________________________________________________________ b) Is the mean IQ of children significantly above the normal IQ of 100? One-tailed test Two-tailed test Ho: _________________________________________________________________ Ha: _________________________________________________________________ c) Does the mean speed of 50 cars as checked by radar on a particular highway has a mean of 65 mph? One-tailed test Two-tailed test Ho: _________________________________________________________________ Ha: _________________________________________________________________ d) Is the mean IQ of the 2nd year UST students below 110? One-tailed test Two-tailed test Ho: _________________________________________________________________ Ha: _________________________________________________________________ e) Are the patients’ mean temperature less than 37ºC? One-tailed test Two-tailed test Ho: _________________________________________________________________ Ha: _________________________________________________________________ TEST of INFERENCE ABOUT THE POPULATION MEAN . If the ideal weight is known to be 60 kg.if the researcher wants to determine if the sample if from the specified population z x or t x s *t-test for single mean use a degree of freedom (df) which is equal to n-1.05. Suppose that a researcher has conducted a study to determine whether the weight of 12-year old boys in his town is 80 lbs. He collected 50 samples with a mean weight of 75 lbs. n n EXAMPLES: 1. Hypotheses: Ho: __________________________________________________________________________________________ Ha: __________________________________________________________________________________________ Test Statistic: Critical Value: ________________________ Decision: _______________ 2. Hypotheses: Ho: __________________________________________________________________________________________ Ha: __________________________________________________________________________________________ Test Statistic: Decision: _______________ Critical Value: ________________________ Conclusion: ______________________________________________________________ . state Ho and Ha and determine whether a one or two-tailed test. can he conclude that the mean weight of the 12year old boys in his town is not 80 lbs? Test at = 0. is the group significantly overweight? Assume = 10 kg and = 0.05.Given the following research questions. xls and as summarized below.01. The results for the 1997 survey are stored in med-time. Hypotheses: Ho: _____________________________________________________________________ Ha: _____________________________________________________________________ Test Statistic: Critical Value: ______________ Decision: _______________ 5.5 hours in internet surfing per day? Test at = 0. A quality control specialist wants to determine if all the capsules being manufactured has a mass of 500 mg. Is there a reason to believe that the 30 Million users in North America spend 6.6.3.16).7847 = 4.3515. A recent survey reveals that there are about 30 Million Internet users in North America. Table 190) Hypotheses: Ho: __________________________________________________________________________________________ Ha: __________________________________________________________________________________________ Test Statistic: Critical Value: ________________________ p-value: _________________ Decision: _______________ 6. the mean diastolic blood pressure in Honolulu Hospital is 73 mmHg with a standard deviation of 11. Conclusion: ______________________________________________________________ Conclusion: _________________________________________ The American Medical Association conducts surveys of its member each year. Results are in internettime.xls (mean = 6. st.dev = 2.0507 = 144 Can we infer that the mean amount of time devoted to patient care per week by all physicians exceeds 45 hours? Test at = 0.9 mg. test whether the mean blood pressure of this group is significantly greater than 70. 2000. A sample of 15 capsules were taken and found a mean of 480 mg with a standard deviation of 27. Suppose that a survey of 200 of these people asked them to report the number of hours they spent per day.05. x s n = 53. Hypotheses: Ho: ___________________________________________________________________________________________ Ha: ___________________________________________________________________________________________ p-value: ___________ Decision: ___________ Conclusion: _____________________________________________________________ . For = 0. Conclusion: ______________________________________________________________ The number of Internet Users is rapidly increasing. From a sample of 25 patients. Can the quality control specialist conclude that the mean mass of all the capsules being manufactured is not 500 mg? Test at 95% confidence interval. A sample of physicians is selected and asked to report the amount of time each devotes to patient care each week.05. Hypotheses: Ho: __________________________________________________________________________________________ Ha: __________________________________________________________________________________________ Test Statistic: Critical Value: ________________________ Decision: _______________ 4. Socioeconomic Characteristics of Medical Practice 1997/1998 and the statistical Abstract of the United States. (Adapted from the American Medical Association.