Hypothesis Testing: One Sample ELEC 412 PROF. SIRIPONG POTISUK General Procedure Although the exact value of a parameter may be unknown, there is often some idea(s) or hypothesi(e)s about its true value Arises as a natural consequence of the scientific method (testing theory against observations) Sample results are used in ascertaining the validity of the hypotheses based on a predetermined criterion 1
General Procedure The decision-making process can be tied to the notion of a confidence interval previously studied. If the resulting CI does not contain the hypothesized value (i.e. 0 ) of the population parameter (i.e., ), the hypothesis that 0 should be rejected. If 0 is within the confidence limits, the hypothesis cannot be rejected. Seven steps in classical hypothesis-testing 2
Step 1: State the Null & Alternative Hypotheses Specifically state the hypotheses to be tested before sampling Assumption to be tested is the null hypothesis H 0 : 0 where 0 is the hypothesized value of the parameter The conclusion accepted contingent on the rejection of H 0, the alternative hypothesis H 1 : 0 or H 1 : > 0 or H 1 : < 0 Selection of H 1 depends on nature of problem Step 2: Select the Level of Significance Establish a criterion to reject H 0 How large does the difference between x and 0 need to be in order to believe 0 not correct? If difference below some predetermined minimum probability level, reject H 0 State the level of risk of erroneously rejecting a true H 0 known as the level of significance and denoted by Type I error The costlier for this type of error, the smaller Fail to reject a false H 0 Type II error 3
Step 3: Determine Distribution of Test Statistic The sample statistic whose value is the basis of the hypothesis-testing decision is called the test statistic Focus again on the standard normal (z), t, and chi-square ( 2 ) distributions Same criteria for choosing the distribution as in CI construction 4
Step 4: Define the Rejection or Critical Region Involve partitioning of the sampling distribution curve according to the significance level The z-, t- or 2 -value used in the partitioning is called the critical value of the test A point at the start or boundary of the rejection region The rejection region equals in total area to the level of significance and is specified as being unlikely to contain the value of the test statistic if H 0 is true. Step 5: State the Decision Rule A decision rule is a formal statement that clearly states the appropriate conclusion to be reached about the null hypothesis based on the value of the test statistic The general format: Reject H 0 in favor of H 1 if the value of the test statistic falls into the rejection region; otherwise, fail to reject H 0. 5
Step 6: Make the Necessary Computations A random sample of items are collected The sample statistic(s) are computed An estimate of the parameter is calculated Step 7: Make a Statistical Decision Make a decision based on the decision rule and the value of the test statistic Managerial decisions vs. Statistical Decisions Statistical results, although objectively determined, shouldn t be blindly accepted; others situational factors must be considered (context of the whole problem) Statistical results do not spell out the course of action to take 6
One-Sample Two-Tailed Tests of Means A two-tailed test is one that rejects the null hypothesis if the test statistic is significantly higher or lower than the hypothesized value of the population parameter. The rejection region has two parts; the total risk of error in rejecting the hypothesis is evenly distributed in each tail Classical Two-Tailed Tests when is known If 0 is not within the CI, X z / 2, then, n assuming a large sample, or normal population 0 X z / 2 or 0 X z / 2 n n X 0 / n z /2 Thus the hypothesized value of 0 is rejected if X 0 / n X 0 or z / 2 / n Z z /2 7
Construction of rejection region with a significant level of.05 Rejection region for a two-tailed test with a significant level of.05 8
With a level of significance of 0.05, the standardized difference between x and 0 becomes significant at +1.96 or -1.96 Classical Two-Tailed Tests when is unknown The correct sampling distribution is the t - distribution if n is 30 or less; otherwise, it is normally distributed In the computation of the test statistic, an estimated standard error must be use instead of the true standard error. For small n, the hypothesized value of 0 is rejected if X s / n 0 T t /2, n 1 9
Example The depth setting on a certain drill press is 2 inches. One could then hypothesized that the average depth of all holes drilled by this machine is = 2 inches. To check this hypothesis (and the accuracy of the depth gauge), a random sample of n = 100 holes drilled by this machine was measured and found to have a sample mean of 2.005 inches with a standard deviation of 0.03 inch. With = 0.05, can the hypothesis be rejected based on these sample data? 10
Example: Statistical Process Control Suppose your summer job includes checking the output of an automatic machine that produces thousands of bolts each hour. This machine, when properly adjusted, makes bolts with a mean diameter of 14.00 mm. Bolts that vary too much in either direction aren t acceptable. It s known from past experience that the diameters are normally distributed about the population mean with = 0.15 mm. If you take a random sample of 6 bolts last hour with diameters: 14.15, 13.85, 13.95, 14.20, 14.30, and 14.35 mm. At the.01 level, does it appear that the machine is properly adjusted? 11
Example: Statistical process Control From the previous example, assume that the is unknown. Again, at the.01 level, does it appear that the machine is properly adjusted? Example A pub owner believes his business sells an average of 17 pints of Ale daily. His partner, thinks his estimate is wrong. A random sample of 36 days shows a mean sales of 15 pints and a sample standard deviation of 4 pints. Test the accuracy of the owner s estimate at the.1 level of significance. 12
Classical One-Tailed Hypothesis Testing If the null hypothesis is not tenable, is the true parameter probably higher or lower than the hypothesized value (but not both)? Null hypothesis H 0 : 0 Alternative hypothesis is one of the following: H 1 : > 0 (right-tailed test) or H 1 : < 0 (left-tailed test) Sometimes, a null hypothesis is enlarged to include an interval of possible values Right-Tailed Tests H 1 : > 0 The rejection region is in the right tail of the sampling distribution The null hypothesis is rejected only if the value of the sample statistic is significantly higher than the hypothesized value Decision Rule: Reject H 0 if z > z or t > t Otherwise, fail to reject H 0 13
Right-Tailed Tests H 1 : > 0 The level of significance ( ) is the total risk of erroneously rejecting H 0 when it s actually true. An area in the right tail is assigned the total risk Left-Tailed Tests H 1 : < 0 The rejection region is in the left tail of the sampling distribution The null hypothesis is rejected only if the value of the sample statistic is significantly lower than the hypothesized value Decision Rule: Reject H 0 if z < z or t < t Otherwise, fail to reject H 0 14
Left-Tailed Tests H 1 : < 0 The level of significance ( ) is the total risk of erroneously rejecting H 0 when it s actually true. An area in the left tail is assigned the total risk Rejection region for a left-tailed test at the.05 level of significance 15
Example A production supervisor at a chemical company wants to be sure that a can of household cleaner is filled with an average of 16 ounces of product. If the mean volume is significantly less than 16 ounces, customers and regulatory agencies will likely complain, prompting undesirable publicity. The physical size of the can doesn t allow a mean volume significantly above 16 ounces. A random sample of 36 cans shows a sample mean of 15.7 ounces. Production records show that is 0.2 ounce. Use this data to conduct a hypothesis test with 0.01 level of significance. 16
Example A vice president for a large corporation claims that the number of service calls on equipment sold by that corporation is no more than 15 per week, on the average. To investigate his claim, service records were checked for n = 36 randomly selected weeks, with sample mean 17 and sample variance 9. Does the sample evidence contradict the vice president s claim at the 5% significance level? 17
Example Muzzle velocities of eight shells tested with a new type of gunpowder yield a sample mean of 2959 feet per second and a standard deviation of s = 39.4. The manufacturer claims that the new gunpowder produces an average velocity of no less than 3000 feet per second. Does the sample provide enough evidence to contradict the manufacturer s claim? Use = 0.05. 18