Type I and Type II Errors Answers

To be considered 18-karat (18K) gold, a piece of jewelry must be made of 75% pure gold. The higher the karats, the more valuable a piece of jewelry. A jewelry designer is purchasing a large quantity of 18K gold from a new supplier. To see if the new supplier is being dishonest about the karat rating in the shipment, the designer melts a random sample of the gold and conducts a hypothesis test with H0: The proportion of metal that is gold is 75%, and Ha: The proportion of metal that is gold is less than 75%. What is a Type I error and its consequence in this context?

To be considered 18-karat (18K) gold, a piece of jewelry must be made of 75% pure gold. The higher the karats, the more valuable a piece of jewelry. A jewelry designer is purchasing a large quantity of 18K gold from a new supplier. To see whether the new supplier is being dishonest about the karat rating in the shipment, the designer melts a random sample of the gold and conducts a hypothesis test with H0: The proportion of metal that is gold is 75%, and Ha: The proportion of metal that is gold is less than 75%. What are a Type I error and a Type II error in this context?

Carlos notices he usually pushes the clear button on his calculator more than once each time he wants to clear the screen. Carlos’ teacher suggests that about 20% of all students have this habit, but Carlos thinks it might be greater. He randomly selects 100 students in his school and finds that 25 of them push the clear button more than once. To determine if these data provide convincing evidence that the proportion of students who push the clear button more than once is greater than 20%, 100 trials of a simulation are conducted. Carlos is testing the hypotheses: H0: p = 20% and Ha: p > 20%, where p = the true proportion of students who push the clear button more than once. Based on the results of the simulation, what is the estimate of the P-value of the test?

A plant in Alamo, TN, manufactures complex transformer components that must meet specific guidelines for safety. One such component is constructed to deliver 1,000 volts of electricity. A component creates a critical safety hazard if it absorbs humidity at a level above 3%. Any components that absorb too much humidity will be destroyed. A quality control inspector uses a random sample of components to conduct a hypothesis test with H0: The humidity level absorbed is 3%, and Ha: The humidity level absorbed is more than 3%. What is a Type I error in this context?

Devon’s tennis coach says that 72% of Devon’s serves are good serves. Devon thinks he has a higher proportion of good serves. To test this, 50 of his serves are randomly selected and 42 of them are good. To determine if these data provide convincing evidence that the proportion of Devon’s serves that are good is greater than 72%, 100 trials of a simulation are conducted. Devon’s hypotheses are: H0: p = 72% and Ha: p > 72%, where p = the true proportion of Devon’s serves that are good. Based on the results of the simulation, what is the estimate of the P-value of the test?

Devon’s tennis coach says that 72% of Devon’s serves are good serves. Devon thinks he has a higher proportion of good serves. To test this, 50 of his serves are randomly selected and 42 of them are good. To determine if these data provide convincing evidence that the proportion of Devon’s serves that are good is greater than 72%, 100 trials of a simulation are conducted. Devon’s hypotheses are: H0: p = 72% and Ha: p > 72%, where p = the true proportion of Devon’s serves that are good. Based on the results of the simulation, the estimated P-value is 0.06. Using = 0.05, what conclusion should Devon reach?

In early 2019, the US rate of recycling plastic water bottles was only 23%. A government agency designs an expensive program to increase the recycling rate. The program will be tested in Texas and, if successful, it will be used nationally. A hypothesis test is conducted with H0: The proportion of water bottles that are recycled is still 23% after the program, and Ha: The proportion of water bottles that are recycled is more than 23% after the program. What is a Type I error and its consequence in this context?

Coach Elyson claims that 90% of her volleyball serves are good serves. The captain of the team thinks Coach Elyson has a lower rate of good serves. To test this, 50 of the coach’s serves are randomly selected and 41 of them are good. To determine if these data provide convincing evidence that the proportion of Coach Elyson’s serves that are good is less than 90%, 100 trials of a simulation are conducted. The team captain’s hypotheses are: H0: p = 90% and Ha: p < 90%, where p = the true proportion of Coach Elyson’s serves that are good. Based on the results of the simulation, what is an estimate of the P-value of the test?

Simone read online that the failure rate in Arizona for the first attempt of the written driver’s test is 60%. Simone thinks the Arizona rate is less than 60%. To investigate, she selects an SRS of 50 Arizona drivers and finds that 27 failed their first written driving test. To determine if this provides convincing evidence that the failure rate for Arizona is less than 60%, 200 trials of a simulation are conducted. Simone’s hypotheses are: H0: p = 60% and Ha: p < 60%, where p = the true proportion of Arizona drivers who fail the first attempt of the written driver’s test. Based on the results of the simulation, the estimated P-value of this test is 0.035. Using a = 0.01, what conclusion should Simone reach?

A restaurant is introducing a new gluten-free recipe for the topping in its baked zucchini recipe. The chef will continue to use this topping if less than 8% of her customers complain about the new taste. Using a random sample of customers, she conducts a hypothesis test with H0: The complaint rate is 8%, and Ha: The complaint rate is less than 8%. What is a Type II error and its consequence in this context?