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Prepared by: Paolo Bautista Preliminaries We wish to test whether a particular assumption/claim regarding the population is true or not. Null Hypothesis (H0) – original assumption Alternative Hypothesis (H1) Determine a critical value to determine whether or not to reject Ho HT : PLBautista Errors Type I Error – reject H0 when in fact it is true Type II Error – fail to reject H0 when it is false Fail to Reject Reject Null Hypothesis is Null Hypothesis is True False Correct Decision Type II Error (β) Type I Error (α) Correct Decision HT : PLBautista Steps in Hypothesis Testing 1. 2. 3. 4. 5. Write the null and alternative hypotheses. Indicate the level of significance. Determine the critical value/s. Compute the test statistic. Decide the conclusion of the test. HT : PLBautista HYPOTHESIS TESTING One Population Two Populations One Mean Case 1 Case 2 One Proportion Difference of Two Means Case 1 to 3 Paired Means Difference of Two One Variance Proportions Difference of Two Variances HT : PLBautista Testing for One Mean Case 1: σ is known, or n ≥ 30 Z-test will be used HT : PLBautista Testing for One Mean The test statistic is given by HT : PLBautista Example 1 A manufacturer of sports equipment has developed a new synthetic fishing line that he claims has a mean breaking strength of 8 kilograms with a standard deviation of 0.5 kilograms. Test the hypothesis that μ = 8 kg against the alternative that μ ≠ 8 kg if a random sample of 50 lines is tested and found to have a mean breaking strength of 7.8 kg. Use a 0.01 level of significance. HT : PLBautista Identify the proper hypotheses The manager of the Danvers-Hilton Resort Hotel stated that the mean guest bill for a weekend is $600 or less. A member of the hotel’s accounting staff noticed that the total charges for guest bills have been increasing in recent months. The accountant will use a sample of weekend guest bills to test the manager’s claim. The manager of an automobile dealership is considering a new bonus plan designed to increase sales volume. Currently, the mean sales volume is 14 automobiles per month. The manager wants to conduct a research study to see whether the new bonus plan increases sales volume. To collect data on the plan, a sample of sales personnel will be allowed to sell under the new bonus plan for a one-month period. HT : PLBautista Identify the proper hypotheses A production line operation is designed to fill cartons with laundry detergent to a mean weight of 32 ounces. A sample of cartons is periodically selected and weighed to determine whether underfilling or overfilling is occurring. If the sample data lead to a conclusion of underfilling or overfilling, the production line will be shut down and adjusted to obtain proper filling. Because of high production-changeover time and costs, a director manufacturing must convince management that a proposed manufacturing method reduces costs before the new method can be implemented. The current production method operates with a mean cost of $220 per hour. A research study will measure the cost of the new method over a sample production period. HT : PLBautista Case 2: σ unknown, AND n < 30 t-test will be used The t-values have n – 1 degrees of freedom The test statistic is given by HT : PLBautista Example 2 Test the hypothesis that the average content of containers of a particular lubricant is 10 liters if the contents of a random sample of 10 containers are 10.2, 9.7, 10.1, 10.3, 10.1, 9.8, 9.9, 10.4, 10.3, and 9.8 liters. Use a 0.01 level of significance and assume that the distribution of contents is normal. HT : PLBautista Testing for One Proportion Z-test will be used The test statistic is given by HT : PLBautista Example 3 A commonly prescribed drug on the market for relieving nervous tension is believed to be only 60% effective. Experimental results with a new drug administered to a random sample of 100 adults who were suffering from nervous tension showed that 70 received relief. Is this sufficient evidence to conclude that the new drug is superior to the one commonly prescribed? Use a 0.05 level of significance. HT : PLBautista Testing for One Variance A chi-square test will be used The test statistic is given by HT : PLBautista Example 4 A manufacturer of car batteries claims that the life of his batteries has a variance equal to 0.81 years. If a random sample of 10 of these batteries have a variance of 1.44 years, is there evidence that the variance exceeds 0.81 a year? Use a 0.05 level of significance HT : PLBautista Testing the Difference of Two Means Case 1: Z-test will be used The test statistic is given by: HT : PLBautista Example 5 A manufacturer claims that the average tensile strength of thread A exceeds the average tensile strength of thread B by less than 12 kilograms. To test this claim, 50 pieces of each type of thread are tested under similar conditions. Type A thread had an average tensile strength of 86.7 kilograms with a standard deviation of 6.28 kilograms, while type B thread had an average tensile strength of 77.8 kilograms with a standard deviation of 5.61 kilograms. Test the manufacturer’s claim using a 0.05 level of significance. HT : PLBautista Case 2: σ1=σ2 unknown, AND n1<30 and n2<30 t-test will be used. The test statistic is given by The degrees of freedom to be used is HT : PLBautista Example 6 A course in mathematics is taught to 12 students by the conventional classroom procedure. A second group of 10 students was given the same course by means of programmed materials. At the end of the semester the same examination was given to each group. The 12 students meeting in the classroom made an average grade of 85 with a standard deviation of 4, while the 10 students using programmed materials made an average of 81 with a standard deviation of 5. Test the hypothesis that the two methods of learning are equal using a 0.10 level of significance. Assume the population to be approximately normal with equal variances. HT : PLBautista Case 3: σ1≠σ2 unknown, AND n1<30 and n2<30 t-test will be used The test statistic is given by The degrees of freedom is given by HT : PLBautista Example 7 An improved manufacturing process is developed. The quality-control tests show that the old process has an average score of 12.8 with a standard deviation of 2.5 based on a sample of 8 observations, while the new process shows an average score of 14.2 with a standard deviation of 1.6 based on a sample of 10 observations. Use a 0.05 level of significance to determine whether there has been a significant increase in the average scores of the new process, assuming unequal variances. HT : PLBautista Paired Observations t-test will be used The test statistic is given by The degrees of freedom to be used is n – 1, where n is the number of pairs HT : PLBautista Example 8 To determine whether membership in a fraternity is beneficial or detrimental to one’s grades, the following grade-point averages were collected over a period of 5 years: Assuming the populations to be normal, test at the 0.05 level of significance whether membership in a fraternity is detrimental to one’s grades. HT : PLBautista Testing for Two Proportions Z-test will be used The test statistic is given by HT : PLBautista Example 9 A vote is to be taken among the residents of a town and the surrounding county to determine whether a civic center will be constructed. To determine if there is a significant difference in the proportion of town voters and county voters favoring the proposal, a poll is taken. If 120 of 200 town voters favor the proposal and 240 of 500 county residents favor it, would you agree that the proportion of town voters favoring the proposal is higher than the proportion of county voters? Use a 0.025 level of significance. HT : PLBautista Testing for Two Variances F-test will be used HT : PLBautista Example 10 Verify if the assumption of equal variance in Example 6 is valid by conducting a test of hypothesis. Use a 0.10 level of significance. HT : PLBautista p-value approach The p-value is the probability that we obtain the sample data, assuming the null hypothesis is true. We reject Ho if the p-value is small. Usually, we use the level of significance as a comparison value. HT : PLBautista Estimation and HT HT : PLBautista