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CBESTA2 EXERCISES – HYPOTHESIS TESTINGTESTS CONCERNING MEANSA 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.A random sample of 100 recorded deaths in the United States during the past year showed an average life span of 71.8 years, with a standard deviation of 8.9 years. Does this seem to indicate that the average life span today is greater than 70 years? Use a 0.05 level of significance.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.The average length time for students to register for fall classes at a certain college has been known to be 50 minutes with a standard deviation of 10 minutes. A new registration procedure using modern computing machines is being implemented. If a random sample of 12 students had an average registration time of 42 minutes with a standard deviation of 11.9 minutes under the new system, test the hypothesis that the population mean is now less than 50, using a level of significance of 0.05. Assume the population of times to be normal.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.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.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:Year12345Fraternity2.02.02.32.12.4Non-fraternity2.21.92.52.32.4Assuming the populations to be normal, test at the 0.05 level of significance whether membership in a fraternity is detrimental to one’s grades.TESTING FOR PROPORTIONSA 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.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.TESTING FOR VARIANCESA 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, do you think that σ2>0.81 a year? Use a 0.05 level of significance.In testing the equality of the population means in Example 4 under Tests Concerning Means, we assumed that the two population variances are equal but unknown. Are we justified in making this assumption? Use a 0.10 level of significance.MORE EXERCISESThe average height of females in the freshman class of a certain college has been 162.5 centimeters with a standard deviation of 6.9 centimeters. Is there reason to believe that there has been a change in the average height if a random sample of 50 females in the present freshman class has an average height of 165.2 centimeters, using a 0.05 level of significance?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.A manufacturer claims that the average tensile strength of thread A exceeds the average tensile strength of thread B by at least 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.A study is made to see if increasing the substrate concentration has an appreciable effect on the velocity of a chemical reaction. With the substrate concentration of 1.5 moles per liter, the reaction was run 15 times with an average velocity of 7.5 micromoles per 30 minutes and a standard deviation of 1.5. With a substrate concentration of 1.0 moles per liter, 12 runs were made yielding an average velocity of 8.8 micromoles per 30 minutes and a sample standard deviation of 1.2. Would you say that the increase in substrate concentration increases the mean velocity by more than 0.5 micromoles per 30 minutes? Use a 0.01 level of significance and assume the populations to be approximately normally distributed with equal variances.The National Association of Home Builders provided data on the cost of the most popular home remodeling projects. Sample data on cost (in thousands of dollars) for the two types of remodeling projects are as follows.Kitchen25.217.422.821.919.723.019.716.921.823.6Master Bedroom18.022.926.424.826.917.824.621.0Determine if there is a significant difference in the average costs for the two types of remodeling projects, using a 0.10 level of significance, and assuming unequal variances.A taxi company is trying to decide whether the use of radial tires instead of belted tires improves fuel economy. Twelve cars were equipped with radial tires and driven over a prescribed test course. Without changing drivers, the same cars were then equipped with regular belted tires and driven once again over the same test course. The gasoline consumption, in kilometers per liter, was recorded as follows:CarKilometers Per LiterRadial TiresBelted Tires14.24.124.74.936.66.247.06.956.76.864.54.475.75.786.05.897.46.9104.94.7116.16.0125.24.9At the 0.025 level of significance, can we conclude that cars equipped with radial tires give better fuel economy than those equipped with belted tires? Assume the populations to be normally distributed.A soft-drink dispensing machine is said to be out of control if the variance of the contents exceeds 1.15 deciliters. If a random sample of 25 drinks from this machine has a variance of 2.03 deciliters, does this indicate at the 0.05 level of significance that the machine is out of control? Assume that the contents are approximately normally distributed.A study is conducted to compute the length of time between men and women to assemble a certain product. Past experience indicates that the distribution of times for both men and women are approximately normal but the variance of the times for women is less than that for men. A random sample of time for 11 men and 14 women produced the following data: the variance for men was 6.1 while the variance for women was 5.3. Test the hypothesis that σ12=σ22 against the alternative σ12>σ22 using a 0.01 level of significance.The gas company claims that two thirds of the houses in a certain city are heated by natural gas. Do we have reason to doubt this claim if, in a random sample of 1000 houses in this city, it is found that 618 are heated by natural gas? Use a 0.02 level of significance. A geneticist is interested in the proportion of males and females in a population that have a certain minor blood disorder. In a random sample of 100 males, 31 are found to be afflicted, whereas only 24 of 100 females tested appear to have the disorder. Can we conclude at the 0.01 level of significance that the proportion of men in the population afflicted with this blood disorder is significantly greater than the proportion of women afflicted? ................
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