Let's approach a problem from two perspectives: An engineer believes a particular component has a fatigue life of 10000. In other words, that component should be expected to fail after 10000 uses. He takes a random sample of 35 components and obtains a sample mean of 9820 with a standard deviation of 2250. Is there any evidence that he is overestimating the fatigue life of the components? (a) What are the null and alternative hypotheses for his test? H0: Se|ect--- a ---Se|ect-- 8 Ha: Se|ect a Se|ect-- a (b) What is the value of the test statistic? (3 decimal places) (c) What is the p-value for his test? (3 decimal places) Now let's examine the problem from another perspective: A particular component has a fatigue life of 10000. An engineer took a random sample of 35 components and tested their fatigue. He obtained a sample mean of 9820 cycles with a standard deviation of 2250. Notice that the standard deviation comes from the sample, so a t distribution would be appropriate to use here. (cl) What is the center of the sampling distribution? (e) How many standard errors from the center of the sampling distribution did the sample mean fall? Use a negative sign if it fell below the center. (3 decimal places) (f) What was the probability of obtaining a sample mean of 9820 or less? (3 decimal places) What could we say about the answers to (c) and (f)? They should be the same. They are unrelated. They should be different. In a survey of pet owners, 37% of the respondents stated that they talk to their pets through phone or video calls. A vet finds this hard to believe, claiming that 37% is too high. Test the hypotheses: Ho: p = 0.37 Ha: p 2000, where ,u is the mean credit card debt for all US college students in 2004. The students in the survey had an average credit card debt of $2169. The p-value for the test was 0.0026. a) Which conclusion is correct for the test using alpha = 0.05. There is insufcient evidence to conclude that college students credit card debt, on average, is greater than $2000. There is sufficient evidence to conclude that college students credit card debt, on average, is greater than $2000. There is insufficient evidence to conclude that college students credit card debt, on average, is about the same as $2000. There is enough evidence to conclude that college students credit card debt, on average, is about the same as $2000. b) If we wanted to reduce the probability of a Type I error, what could we do? Decrease our significance level a We cannot affect the probability of a type I error. Increase our significance level a Suppose we test Ho: p = 0 versus Ha: y > 0, and get a P-value of 0.02. (a) What would the decision be for a significance level of 0.05? Interpret in context. Reject the null hypothesis. There is sufficient evidence that the mean is greater than 0. Reject the null hypothesis. There is insufficient evidence that the mean is greater than 0. Fail to reject the null hypothesis. There is sufficient evidence that the mean is greater than 0. Fail to reject the null hypothesis. There is insufficient evidence that the mean is greater than 0. (b) If the decision in (a) is in error, what type of error is it? Type I Type II (c) Suppose the significance level were instead 0.01. What decision would you make, and if it is in error, what type of error would it be? Reject the hull with a possible Type I error. Reject the hull with a possible Type 11 error. Fail to reject the null with a possible Type I error. Fail to reject the null with a possible Type II error. Ten years ago, the average amount of time a family spent eating supper together was 27.8 minutes. A sociologist believes this average has decreased. She randomly selects a sample of 12 families and finds the sample average to be 24.307 minutes. The sample data look about bell-shaped. She calculated a test statistic oft = -1.461. (a) What is the p-value ? (3 decimal places) (b) If she conducts the test at signicance level 0.05, what should the conclusion be? Do not reject the null hypothesis. There is evidence to indicate that the average time has decreased at significance level 0.05. Reject the null hypothesis. There is not enough evidence to indicate that the average time has decreased at significance level 0.05. Reject the null hypothesis. There is evidence to indicate that the average time has decreased at significance level 0.05. Do not reject the null hypothesis. There is not enough evidence to indicate that the average time has decreased at significance level 0.05. (c) If she conducts the test at significance level 0.10, what should the conclusion be? Do not reject the null hypothesis. There is evidence to indicate that the average time has decreased at significance level 0.10. Do not reject the null hypothesis. There is not enough evidence to indicate that the average time has decreased at significance level 0.10. Reject the null hypothesis. There is not enough evidence to indicate that the average time has decreased at significance level 0.10. Reject the null hypothesis. There is evidence to indicate that the average time has decreased at significance level 0.10. A store owner claims the average age of her customers is 25 years. She took a survey of 33 randomly selected customers and found the average age to be 27.6 years with a standard error of 1.160. Carry out a hypothesis test to determine if her claim is valid. (a) Which hypotheses should be tested? H0: [,1 = 27.6 vs. Ha: )4 27.6 Ho: M = 25vs. Ha: y > 25 H0: p = 25 vs. Ha: p 25 Ho: M = 25vs. Ha: Mai: 25 (b) Find the test statistic: (Use 4 decimals.) (c) What is the P-value? (Use 4 decimals.) (d) What should the store owner conclude, for a = 0.05? Reject the initial claim of 25 years. There is sufficient evidence the mean customer age is different than 25. Reject the initial claim of 25 years. There is insufficient evidence the mean customer age is different than 25. Do not reject the initial claim of 25 years. There is insufficient evidence the mean customer age is different than 25. Do not reject the initial claim of 25 years. There is sufficient evidence the mean customer age is different than 25. (e) If mean customer age really is equal to 25 years, but you conclude it is different than 25 years, which type of error did you make, if any? Type I error Type II error The p-value is correct; therefore no error was made Both Type I and Type II error In the 19905, it was generally believed that genetic abnormalities affected about 7% of a large nation's children. Some people believe that the increase in the number of chemicals in the environment has led to an increase in the incidence of abnormalities, which could have important implications for health insurance companies. A recent study examined 354 randomly selected children and found that 32 of them showed signs of a genetic abnormality. (a) Which hypotheses should be used to test if the proportion of genetic abnormalities has increased in recent years? H0: p = 0.07 vs. Ha: p at 0.07 H0: p = 0.0904 vs. Ha: p 0.0904 H0: p = 0.07 vs. Ha: p > 0.07 (b) Are the conditions met for doing the hypothesis test? The 15 successes and failures condition is met. The sample was randomly chosen. The children taking part in the study were independent of each other. None of the conditions are met. (c) What is the p-value? (Use 3 decimals.) (d) What does this p-value mean? The p-value is the chance of observing 32 or more children with genetic abnormalities in a random sample of 354 children, if the true proportion of children with genetic abnormalities is 7%. The p-value is the chance of observing 32 or more children with genetic abnormalities in a random sample of 354 children, if the true proportion of children with genetic abnormalities is 9.04%. The p-value is the chance of observing 7% of children with genetic abnormalities. The p-value gives the actual percentage of children who have genetic abnormalities. (e) What is the conclusion of the hypothesis test, for a = 0.05? Do not reject H0. There is insufficient evidence that more than 7% of this nation's children have genetic abnormalities. Do not reject H0. There is sufficient evidence that more than 7% of this nation's children have genetic abnormalities. Reject H0. There is sufficient evidence that more than 7% of this nation's children have genetic abnormalities. Reject H0. There is insufficient evidence that more than 7% of this nation's children have genetic abnormalities. (f) Do this study show that environmental chemicals cause congenital abnormalities? Yes, the hypothesis test shows that environmental chemicals cause the genetic abnormality. This study does not address what causes the genetic abnormality. No, the hypothesis test shows that environmental chemicals do not cause the genetic abnormality. An industrial plant claims to discharge no more than 1000 gallons of wastewater per hour, on the average, into a neighboring lake. An environmental action group decides to monitor the plant, in case this limit is being exceeded. A random sample of 42 hours is selected over a period of a month. The data are unimodal and roughly symmetric. E: 1053, S: 698, and Standard Error = 107.704 (a) Using StatCrunch, the p-value = (3 decimal places) (b) Are the conditions met to conduct this test? Yes, np and n(l-p) are both > 15. Yes, because the data are approximately normal. 0 Yes, the population is approximately normal because n > 30. No, np and n(l-p) are not both > 15. (c) At alpha = 0.05, which of the following is the correct conclusion? 0 Reject the null hypothesis. There is sufficient evidence that the wastewater discharged by the industrial plant exceeds 1000 gallons per houn Reject the null hypothesis. There is insufficient evidence that the wastewater discharged by the industrial plant exceeds 1000 gallons per houn Do not reject the null hypothesis. There is evidence that the wastewater discharged by the industrial plant exceeds 1000 gallons per houn Do not reject the null hypothesis. There is insufficient evidence that the wastewater discharged by the industrial plant exceeds 1000 gallons per hour