CHAPTER 8 Hypothesis Testing Student Notes Stat 200 Elementary Statistics Introduction Hypothesis testing In hypothesis testing the researcher must define 1 2 3 4 5 6 7 Methods to Test Hypotheses 1 2 3 Section 8 1 Steps in Hypothesis Testing Traditional Method statistical hypothesis is a conjecture about a population parameter which may or may not be true There are two types of statistical hypotheses for each situation A 1 2 Null hypothesis symbolized by is a statistical hypothesis that states that there is no difference between a parameter and a specific value or that there is no difference between two parameters Alternative hypothesis symbolized by is a statistical hypothesis that states the existence of a difference between a parameter and a specific value or states that there is a difference between two parameters Stating the Null and Alternative Hypothesis Situation A Will the pulse rate of a patient increase decrease or remain unchanged after a certain medication is taken The mean pulse rate of the population is 82 beats per minute Situation B An additive is to increase the life of an automobile battery If the mean lifetime of the battery without the additive is 36 months then the hypotheses are Situation C A contractor wishes to lower heating bills by using a special type of insulation in houses If the average of the monthly heating bills is 78 her hypotheses about heating costs with the use of insulation are Example 1 State the null and alternative hypotheses for each a A researcher thinks that if expectant mothers use vitamin pills the birth weight of the babies will increase The average birth weight of the population is 8 6 pounds b An engineer hypothesizes that the mean number of defeats can be decreased in a manufacturing process of compact disks by using robots instead of humans for certain tasks The mean number of defective disks per 1000 is 18 c A psychologist feels that playing soft music during a test will change the results of the test The psychologist is not sure whether the grades will be higher or lower In the past the mean of the scores was 73 Statistical test Test value Possible Outcomes of a Hypothesis Test Summary of Possible Outcomes Type I error Type II error and Probabilities Critical Values critical or rejection region is the range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected The noncritical or nonrejection region is the range of values of the test value that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected The One Tailed Test Two Tailed Test Example 2 Find the critical value s for each situation and draw the appropriate figure a A left tailed test with a 0 10 b A two tailed test with a 0 02 c A right tailed test with a 0 005 Hypothesis Testing Traditional Method 1 2 3 4 5 Section 8 2 Z Test for a Mean z test Example 1 A researcher reports that the average salary of assistant professors is more than 42 000 A sample of 30 assistant professors has a mean salary of 43 260 At a 0 05 test the claim that assistant professors earn more than 42 000 a year The standard deviation of the population is 5230 Example 2 A researcher claims that the average cost of men s athletic shoes is less than 80 He selects a random sample of 36 pairs of shoes from the catalog and finds the following costs Is there enough evidence to support the researcher s claim at a 0 10 60 70 75 55 80 55 50 40 80 70 50 95 120 90 75 85 80 60 110 65 80 85 85 45 75 60 90 90 60 95 110 85 45 90 70 70 Example 3 A Medical Foundation reports that the average cost of rehabilitation for stroke victims is 24 672 To see if the average cost of rehabilitation is different at a particular hospital a researcher selects a random sample of 35 stroke victims at the hospital and finds that the average cost of their rehabilitation is 25 226 The standard deviation of the population is 3251 At a 0 05 can it be concluded that the average cost of stroke rehabilitation at a particular hospital is different from 24 672 Outcomes of a Hypothesis Testing I Claim is H0 III Claim is H1 P value Decision Rule When Using a P Value Example 4 A researcher wishes to test the claim that the average are of lifeguards in Ocean City is greater than 24 years She selects a sample of 36 guards and finds the mean of the sample to be 24 7 years with a standard deviation f 2 years Is here evidence to support the claim at a 0 05 Use the P value method Example 5 A researcher claims that the average wind speed in a certain city is 8 miles per hour A sample of 32 days has an average wind speed of 8 2 miles per hour The standard deviation of the sample is 0 6 mile per hour At a 0 05 is there enough evidence to support the claim Use the P value method Section 8 3 t test for a Mean t test The formula for the t test is t The degrees of freedom are d f n 1 Example 1 X s n a Find the critical t value for a 0 05 with d f 16 for a right tailed t test b Find the critical t value for a 0 01 with d f 22 for a left tailed t test c Find the critical t value for a 0 10 with d f 18 for a two tailed t test d Find the critical t value for a 0 05 with d f 28 for a right tailed t test Example 2 A job placement director claims that the average starting salary for nurses is 24 000 A sample of 10 nurse s salaries has a mean of 23 450 and a standard deviation of 400 Is there enough evidence to reject the director s claim at a 0 05 Step 1 Step 2 Step 3 Step 4 Step 5 Example 3 An educator claims that the average salary of substitute teachers in a certain school district is less than 60 per day A random sample of eight districts is selected and the daily salaries are shown Is there enough evidence to support the educator s claim at a 0 10 60 56 60 55 70 55 60 55 Step 1 Step2 Step 3 Step 4 Step 5 Example 4 Find the P value when the t test value is 2 056 the sample size is 11 and the test is right tailed Example 5 Find the P value when the t test value is 2 983 the sample size is 6 and the test is two tailed Example 6 A physician claims that joggers maximal volume oxygen uptake is greater …