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Name: ____________________________________Hypothesis WS #11) Determine if each pair is a set of legitimate hypotheses. If it’s not legitimate, explain why.a) H0: p.4: Ha: p = .4b) H0: = 16: Ha: > 16c) H0: = 24: Ha: > 242) For each situation, state the null and alternative hypotheses. Be sure to define parameters in context!a) U.S. children have a mean blood cholesterol level of 170. Researchers have postulated that, due to differences in diet, Japanese children have a lower mean blood cholesterol level than U.S. children.b) A water quality control board reports that water is unsafe for drinking if the nitrate concentration exceeds 0.03%. Water specimens are taken at random from a well.c) Last year, your company’s service technicians took an average of 2.6 hours to respond to trouble calls from business customers who had purchased service contracts. Do this year’s data show a different average response time?d) Census data show that the proportion of consumers in the area served by a shopping mall with incomes over $50,000 is 68%. A market research firm questions random shoppers at the mall. The researchers suspect that the proportion of shoppers with incomes over $50,000 is higher than the proportion of the area population.e) The diameter of a spindle in a small motor is supposed to be 5 mm. If the spindle is either too small or too large, the motor will not work properly. The manufacturer measures the diameter in a random sample of the motors to determine whether the mean diameter has moved away from the target.f) Last season, the Detroit Tigers scored an average of 3.4 runs per game. After the first three weeks of the new season, a Tigers commentator believes that, based on the games he’s seen so far, the Tigers will score more runs per game on average this year.(continued on back)3) Check the conditions for a hypothesis test in each scenario.a) A pharmaceutical manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The hardness of a sample from each lot of tablets produced is measured in order to control the compression process. The hardness data for a random sample of 20 tablets are:11.62711.61311.49311.60211.36011.37411.59211.45811.55211.46311.38311.71511.48511.50911.42911.47711.57011.62311.47211.531b) In a statewide survey of Michigan residents, 46% say they are fans of the University of Michigan. To test whether the proportion of UM fans in Grand Blanc differs from the statewide proportion, you take a random sample of 100 Grand Blanc residents, and find that 39% of them say they are UM fans.c) Bottles of a popular cola are supposed to contain 300 milliliters of cola. There is some variation from bottle to bottle because the filling machinery is not perfectly precise. In a random sample of 50 bottles, the mean amount is 299.2 milliliters with standard deviation 4.6 milliliters.Name: ____________________________________Hypothesis WS #2For each of the following p-values, state whether you would reject or fail to reject H0 for the given level.1) p-value = 0.234 when = 0.052) p-value = 0.024 when = 0.053) p-value = 0.024 when = 0.014) p-value = 0.024 when = 0.10For each of the following tests, a) draw & shade the curve for H0: p = .25 b) calculate the p-value of the given test statistic c) write the appropriate conclusion5) right-tail test z = 2.056) left-tail test z = -1.957) two-tailed test z = 1.758) two-tailed test z = -2.059) Direct mail advertisers send solicitations (junk mail) to thousands of potential customers in the hope that some will buy their product. The mailing will be cost-effective if at least 13% of recipients purchase the product. A company wants to test customers’ response to a new flyer, since they’re worried that not enough people are purchasing the product. They send the flyer to 1000 randomly selected people and get 123 purchases.a) Write the hypotheses the company is testing.b) The test statistic for this sample is -.658. Calculate the p-value and write the appropriate conclusion in context.Name: ____________________________________Hypothesis WS #31) A magazine is planning the launch of an online edition, but is worried not enough readers would subscribe. The magazine plans to scrap its plans if it’s convinced that less than 30% of current readers would subscribe. The magazine contacts a random sample of 500 current subscribers and 137 of those surveyed expressed an interest. Is this sufficient evidence for the magazine to start the online edition?2) During the 2000 season, the home team won 138 of the 240 regular season National Football League games. Is this strong evidence of a home field advantage (home teams winning more than half the games)?(continued on back)3) In a rural area, only about 30% of the wells that are drilled find adequate water at a depth of 100 feet or less. A local man claims to be able to find water by “dowsing” (using a forked stick to indicate water). Twenty-seven out of 80 of his customers have wells less than 100 feet deep. Does this evidence suggest that the proportion of wells found with depths of 100 feet or less is different than the typical 30%?4) John Wayne died of cancer. In 1955, he was in Utah shooting the film The Conqueror. Across the state line, in Nevada, the U.S. military was testing the atomic bomb. Radioactive fallout drifted across the film location. A total of 46 of the 220 people working on the film eventually died of cancer. Is there sufficient evidence that the death rate observed in this movie crew is higher than the national proportion of 13.6%?Name: ____________________________________Hypothesis WS #4Perform an appropriate hypothesis test to answer each question. Don’t forget to check the conditions, define parameters in the hypotheses, and write your conclusion in context.59372501085851) A financial report indicates that for the past decade, the cattle ranching industry in Colorado has held steady with a mean total cash receipt of about 2.7 billion dollars annually. A random sample of 30 ranches has a mean total cash receipt of 2.85 billion dollars with standard deviation of 0.55 billion dollars. Is the Colorado cattle ranching industry expanding?2) The Statistical Abstract of the United States (109th edition) reported that the average cost per day of owning an automobile in the United States is $7.62 with standard deviation of $1.77. This includes the cost of the car, general maintenance, gasoline, and insurance. A random sample of 54 students who own cars had an average cost per day of $6.78. Is a student’s average daily expense less than the national average? (continued on back)59372501225553) The Statistical Abstract of the United States (109th edition) reported that the mean age of all U.S. registered merchant vessels is 23.6 years. Records for 38 randomly selected U.S. registered grain cargo ships showed that the mean age was 26.7 years with a standard deviation of 5.2 years. Is this type of ship older than the national average of all merchant vessels?Name: ____________________________________Hypothesis WS #5Perform an appropriate hypothesis test to answer each question. Don’t forget to check the conditions, define parameters in the hypotheses, and write your conclusion in context.1) To ensure a safe workplace, it is important that workers not be asked to perform tasks, such as lifting, that exceed their capabilities. The following data on maximum weight of lift (in kg) for a frequency of 4 lifts per minute is from a random sample of males, ages 18-30: 60559952286025.836.626.321.827.223.331.223.5Do the data suggest that the population mean maximum weight of lift exceeds 25 kg?2) Based on information from State Farm Insurance Company, 67% of all damage liability claims are made by single people under the age of 25. A random sample of 53 claims in Genesee County showed 42 claims were made by single people under the age of 25. Does this indicate that the percent of insurance claims by single people under the age of 25 in Genesee County is higher than the national percent?(continued on back)3) Much concern has been expressed in recent years regarding the practice of using nitrates as meat preservatives. In one study of the possible effects of these chemicals, bacteria cultures were grown in a medium containing nitrates. The rate of uptake of radio-labeled amino acid was then determined for each culture, yielding the following observations: 72516871963268669094584989577978706478947883817875238727 7468Suppose that it is known that the true average uptake for cultures without nitrates is 8000. Do the data suggest that the addition of nitrates results in a decrease in the true average uptake?4. A large university provides housing for 10 percent of its graduate students to live on campus. The university’s housing office thinks that the percentage of graduate students looking for housing on campus may be more than 10 percent. The housing office decides to survey a random sample of graduate students, and 62 of the 481 respondents say that they are looking for housing on campus.On the basis of the survey data, would you recommend that the housing office consider increasing the amount of housing on campus available to graduate students? Give appropriate statistical evidence to support your recommendation.Name: ____________________________________Comparing Intervals & Tests2004 #6A pharmaceutical company has developed a new drug to reduce cholesterol. A regulatory agency will recommend the new drug for use if there is convincing evidence that the mean reduction in cholesterol level after one month of use is more than 20 milligrams/deciliter (mg/dl), because a mean reduction of this magnitude would be greater than the mean reduction for the current most widely used drug.The pharmaceutical company collected data by giving the new drug to a random sample of 50 people from the population of people with high cholesterol. The reduction in cholesterol level after one month of use was recorded for each individual in the sample, resulting in a sample mean reduction and standard deviation of 24 mg/dl and 15 mg/dl, respectively.a) The regulatory agency decides to use an interval estimate for the population mean reduction in cholesterol level for the new drug. Provide this 95 percent confidence interval. Be sure to interpret this interval.b) Because the 95 percent confidence interval includes 20, the regulatory agency is not convinced that the new drug is better than the current best-seller. The pharmaceutical company tested the following hypotheses.H0: = 20 versus Ha: > 20,where represents the population mean reduction in cholesterol level for the new drug.The test procedure resulted in a t-value of 1.89 and a p-value of 0.033. Because the p-value was less than 0.05, the company believes that there is convincing evidence that the mean reduction in cholesterol level for the new drug is more than 20. Explain why the confidence interval and the hypothesis test led to different conclusions.1) A forester wishes to estimate the mean growth of seedlings in a large timber plot since last year. A random sample of 100 seedlings is taken and it is found that the one-year growth yields an average of 12.8 cm with a standard deviation of 2.5 cm.a) Find a 95% confidence interval for the true mean growth.b) What significance level corresponds to a 95% confidence level?c) At the significance level you identified in part (b), does the interval in part (a) lead you to reject or fail to reject the H0? Justify your answer.(continued on back)2) In a random sample of 250 high school students, it was found that 84% of them ultimately graduated.a) Find a 98% confidence interval for the true proportion of all high school students who graduate.b) What significance level corresponds to a 98% confidence level?c) At the significance level you identified in part (b), does the interval in part (a) lead you to reject or fail to reject the H0? Justify your answer.3) The Rocky Mountain News (January 24, 1994) indicated that the 20-year mean snowfall in the Denver/Boulder region is 28.76 inches with standard deviation of 7.5 inches. However, for the winter of 1993-1994, the average snowfall for a sample of 32 different locations was 33 inches.a) Perform a hypothesis test ( = .05) to determine if this data indicates that the average snowfall for the 1993-1994 winter was higher than the previous 20-year average. (Don’t worry about checking conditions.)b) What confidence level corresponds to the significance level in part (a)?c) Suppose you constructed a confidence interval using the confidence level in part (b). Based on the result of your hypothesis test in part (a), would the 20-year mean of 28.76 fall within your interval? Justify your answer.4) To assess the impact of quality circles (groups of employees who meet to discuss issues related to their work) on employee job satisfaction, 73 employees who participated in quality circles were studied. Based on previous surveys, the proportion of employees whose job satisfaction improves over time is 68%. Out of the 73 employees who experienced quality circles, 55 had their job satisfaction improve.a) At the 10% significance level, is the proportion of employees participating in quality circles whose satisfaction improved higher than that of the general working population?b) What confidence level corresponds to a 10% significance level?c) Suppose you constructed a confidence interval using the confidence level in part (b). Based on the result of your hypothesis test in part (a), would the population proportion of 68% fall within your interval? Justify your answer.Name: ____________________________________Matched Pairs WS1) Elite Foods, a supermarket, is losing customers and suspects it’s because another neighborhood market has lower prices. To determine whether this is the case, researchers from Elite Foods take a random sample of ten items from their store and compare their prices to prices for the same items at the other store. Compute and interpret a 95% confidence interval for the true mean difference in prices between the stores.Item 1 2 3 4 5 6 7 8 9 10Elite Foods1.652.191.993.49.991.592.894.501.191.99Other Store1.492.002.092.99.991.792.394.25.991.792) Based on your confidence interval in #1, is there evidence that the other store’s prices are lower? Explain.3) A school wishes to compare the effectiveness of two math curricula it’s developed. School researchers take two random samples of final math exam scores: one from students who studied the first curriculum and one from students who studied the second curriculum. Researchers then pair these scores based on grade point averages, grade level, and after school activities. Is there sufficient evidence to suggest that there is a difference between the final exam scores from the two different curricula?Student CurriculumPairAB183902909437875482785807269394784808909698384108688117982129696139390148684157983168384178384188587198085207679 Name: ____________________________________Errors WS #1For each of the following: a) State the null & alternative hypotheses. b) State the type I & type II errors and their consequences. c) Decide if one error is more serious, and explain why.1) An Apple subsidiary has created a new manufacturing method for producing iPads, which they claim will reduce the time necessary for assembling the parts. On average, it currently takes 75 minutes to produce an iPad. The retooling of the manufacturing plant for this change is very expensive and will involve a lot of downtime, so Apple must decide if the new manufacturing process will truly reduce assembly time.2) You are interested in building a new restaurant in an area – but only if the income level is high enough to support the business. In order to open the restaurant, you estimate that the per capita income must be higher than $45,000.3) A manufacturer of handheld calculators receives very large shipments of printed circuits from a supplier. It is too costly and time-consuming to inspect all incoming circuits, so when each shipment arrives, a sample is selected for inspection. If more than 5% of the circuits are defective, the entire shipment is returned to the supplier due to inferior quality.Name: ____________________________________Errors WS #21) A medical researcher has tested a new treatment for poison ivy against the traditional ointment. She wanted to know if the new treatment alleviated a larger proportion of the poison ivy rash than the traditional ointment, which typically alleviates about 85% of the rash. With a p-value of 0.047, she concludes the new treatment is more effective. State the hypotheses she tested.What does the p-value of 0.047 mean in context?Which error could she have potentially made? What is a consequence of this error?2) Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find that the proportion of items that must be rejected is higher than 5%, they will halt the assembly process until the problem can be identified and repaired.State the hypotheses the managers are testing.In this context, what is a type I error? What is a consequence of this error?In this context, what is a type II error? What is a consequence of this error?Which type of error would the factory owner probably consider more serious? Why?Which type of error might customers consider more serious? Why?Their test currently uses a 5% level of significance. How would changing to a significance level of 1% affect the probability of a type I error occurring?How would changing to a significance level of 1% affect the probability of a type II error occurring?3) Highway safety engineers test new signs, hoping that increased reflectivity will make them more visible to drivers. Volunteers drive through a test course with several of the new and old style signs and rate which kind shows up the best.Is the test of the new signs a one-tailed or two-tailed test? Why?In this context, state the type I & type II errors. What are the consequences of each?Which error do you think is more serious?Based on your answer to part (c), which level of might you choose? Why?The engineers hoped to base their decision on the reactions of 50 drivers, but time and budget constraints may force them to cut back to 20. How would this affect the probability of type I and type II errors?Mixed Inference Review4) The Environmental Protection Agency sets limits on the maximum allowable concentration of certain chemicals in drinking water. For the substance PCB (polychlorinated biphenyl, a toxic compound used in coolant fluids in many transformers, capacitors, and electric motors), the limit is set at 5 ppm (parts per million). A random sample of 36 water specimens from a well results in a mean PCB concentration of 5.2 ppm and a standard deviation of 0.6 ppm. Does this data indicate that the water is unsafe to drink at the .01 significance level?5) Identify the type I & II errors for the hypotheses in question 4. State a consequence for each. Which is more serious?6) According to government data, 22% of American children under the age of 6 live in households with incomes below the official poverty level. A study of childhood learning chooses an SRS of 300 children under age 6 from a large city and finds that 80 live in households with incomes below the official poverty level. Is there significant evidence that a higher proportion of children in this city are living in poverty than the national proportion?7) Below is the average temperature (in Fahrenheit) during each month of the year for Buffalo, NY and Grand Rapids, MI (according to the U.S. Department of Commerce Environmental Data Service). Do these data indicate that there is a significant difference in average temperatures between the two cities?MonthJFMAMJJASONDBuffalo25.124.532.343.354.664.770.368.962.651.840.029.5Gr. Rapids24.424.433.946.557.968.072.670.663.252.139.328.5Name: ____________________________________Errors WS #3For questions 1 & 2, identify whether the statements are correct decisions (C) or errors (I or II). For correct decisions, also state whether this indicates the power (P) of the test.1) A researcher tests whether the mean cholesterol level among those who eat frozen pizza exceeds the value considered to indicate a health risk. (H0: mean level is OK vs. Ha: mean level is too high)_____a) The researcher decides that the mean cholesterol level among those who eat frozen pizza exceeds the value considered to indicate a health risk, when in fact the mean cholesterol level really is higher._____ b) The researcher decides that the mean cholesterol level among those who eat frozen pizza exceeds the value considered to indicate a health risk, when in fact the mean cholesterol level really is not higher._____c) The researcher decides that the mean cholesterol level among those who eat frozen pizza does not exceed the value considered to indicate a health risk, when in fact the mean cholesterol level really is higher.2) A cross-country skier is trying to decide whether to use a new racing wax on his skis. The wax would be worth the additional price if he could, on average, complete a known course in less than his usual time.(H0: mean time is the same with new wax vs. Ha: mean time is less with new wax) _____a) The skier decides not to use the new wax, when in fact the new wax would help him race faster._____b) The skier decides to use the new wax, when in fact the new wax does not help him race faster._____ c) The skier decides not to use the new wax, when in fact the new wax does not help him race faster.3) Fill in the chart by indicating whether each value increases or decreases. If you increase…P(Type I Error)P(Type II Error)Powerndistance between 0 and a4) Which of the following is not true?A way to reduce the chance of a Type II error is to increase sample size.If sample size remains constant, then reducing will increase the value of . depends on the null hypothesis; depends on the null hypothesis and the value of an alternative mean.For the hypothesis test of a parameter, must equal .All of these are correct.5) Given H0: = 30, Ha: < 30, if you conclude that the mean is less than 30 when it is actually 27, thenyou have made a Type I erroryou have made a Type II errorthe result of you test was not significantyou have drawn a correct conclusionyou failed to reject the null hypothesis6) Given = 0.05, which of the following is true? = 0.95The power of the test is 0.95. = P(rejecting H0 when H0 is true) = P(rejecting H0 when H0 is false)The value of is independent of the value of .Name: ____________________________________Mixed Inference Review1) Are women marrying at a later age? A report stated that in 1970 (according to census data), the mean age of brides marrying for the first time was 20.8 years. In 2010, a random sample of 100 brides had a mean age of 23.9 with a standard deviation of 6.4 years. Is there sufficient evidence to support the claim that in 2010, women were marrying later in life than in 1970?2) The Wall Street Journal stated that 45% of American households repair a vacuum cleaner annually. In a random sample of 25 households in Michigan, the 10 repaired a vacuum cleaner in a given year. Does this data indicate that the proportion of Michigan households who repair a vacuum cleaner in a year is different than the national proportion?3) The red blood cell count (RBC) in millions per cubic millimeter of whole blood for healthy adults is normally distributed with a mean of 4.8 and a standard deviation of 0.4. An adult patient has taken six blood tests over the past several months with a mean RBC of 4.47.a) Construct and interpret a 95% confidence interval for this patient’s true mean RBC.b) Based on this interval, does this indicate that the patient’s RBC is lower than the population mean? Why?4) The American Meteorological Society has a rating system to classify Nor’easter storms that frequently hit New England states and can cause much damage near ocean coasts. A “severe” storm brings waves with an average peak height of 16.4 feet and a standard deviation of 3.2 feet. When a Nor’easter occurs, the AMA must decide whether to classify it as “severe.”Suppose that a Nor’easter is in progress and has been given a “severe” storm rating. A random reading of 36 peak waves showed an average wave height of 15.4 feet. Perform an appropriate hypothesis test to determine if the storm’s rating should be downgraded.5) Consumer Reports gave the following data on the life (in hours) of AA Duracell batteries for a certain toy. Compute a 98% confidence interval for the mean life of all AA Duracell batteries for that toy.2.32.54.26.15.75.51.31.55.45.31.81.95.21.85.16) You want to estimate the percentage of voters who will vote for Senator Porkbelly in the next election to within 2% with 95% confidence. What is the minimum sample size needed to do this?7) The average hemoglobin count (HC) for healthy adult women is 14 grams per 100 milliliters of whole blood. Suppose that a female patient has taken 12 laboratory blood samples during the past year. The HC counts sent to her doctor are listed below. Does the information indicate that this patient has significantly higher HC than the population?1914232015192116181816218) For their total production of light bulbs, the management of an electronics firm knows that = 150 hours. How large a sample is needed if they want to be 90% confident that the estimate of the average production time will be within 30 hours of the correct value?9) Vitamin D, whether ingested as a dietary supplement or produced naturally when sunlight falls upon the skin, is essential for strong, healthy bones. A recent study of 2700 children randomly selected from all parts of England found 20% of them deficient in vitamin D. Compute a 96% confidence interval for the proportion of children in England with Vitamin D deficiency. ................
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