Answers CA-Statistics and Probability BIntroduction to Hypothesis Testing

Type I and Type II Errors Answers

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Lucy recently asked the servers at her restaurant to only give straws to customers who request them. She thinks that about half of the customers will ask for straws but hopes that the rate will be less than half. She randomly selects 100 customers and finds that 43 of them ask for a straw. To determine if these data provide convincing evidence that the proportion of customers who will ask for a straw is less than 50%, 150 trials of a simulation are conducted. Lucy is testing the hypotheses: H0: p = 50% and Ha: p < 50%, where p = the true proportion of customers who will ask for a straw. Based on the results of the simulation, the estimated P-value is 0.0733. Using a = 0.05, what conclusion should Lucy reach?

Because the P-value of 0.0733 > a, Lucy should reject Ha. There is convincing evidence that the proportion of customers who will ask for a straw is less than 50%.

Because the P-value of 0.0733 > a, Lucy should reject Ha. There is not convincing evidence that the proportion of customers who will ask for a straw is less than 50%.

Because the P-value of 0.0733 > a, Lucy should fail to reject H0. There is convincing evidence that the proportion of customers who will ask for a straw is less than 50%.

Because the P-value of 0.0733 > a, Lucy should fail to reject H0. There is not convincing evidence that the proportion of customers who will ask for a straw is less than 50%.

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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?

Because the P-value of 0.06 > a, Devon should reject Ha. There is convincing evidence that the proportion of serves that are good is more than 72%.

Because the P-value of 0.06 > a, Devon should reject Ha. There is not convincing evidence that the proportion of serves that are good is more than 72%.

Because the P-value of 0.06 > a, Devon should fail to reject H0. There is convincing evidence that the proportion of serves that are good is more than 72%.

Because the P-value of 0.06 > a, Devon should fail to reject H0. There is not convincing evidence that the proportion of serves that are good is more than 72%.

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?

11%

17%

20%

25%

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?

Type I error: The gold shipment truly is made of 75% gold, but the designer concludes that it is made of less than 75% gold.Type II error: The gold shipment truly is made of less than 75% gold, but the designer concludes that it is made of 75% gold.

Type I error: The gold shipment truly is made of 75% gold, but the designer concludes that it is made of less than 75% gold.Type II error: The gold shipment truly is made of less than 75% gold, and the designer concludes that it is made of less than 75% gold.

Type I error: The gold shipment truly is made of 75% gold, and the designer concludes that it is made of 75% gold.Type II error: The gold shipment truly is made of less than 75% gold, but the designer concludes that it is made of 75% gold.

Type I error: The gold shipment truly is made of less than 75% gold, but the designer concludes that it is made of 75% gold.Type II error: The gold shipment truly is made of 75% gold, but the designer concludes that it is made of less than 75% gold.

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?

98%

Animal shelters in a county need at least 15% of their animals to be adopted weekly to have room for the new animals that are brought into the various shelters. The county manager takes a random sample of shelters each week to estimate the overall proportion of animals that are adopted. If he concludes that the proportion has dropped below 15%, he will not accept any new animals into the shelters that week. He tests the hypotheses: H0: The adoption rate is 15%, and Ha: The adoption rate is less than 15%. What is a Type II error, and what is its consequence in this context?

The manager believes the adoption rate is still 15%, when it actually has dropped below 15%. The manager will accept more animals into the shelters and will run out of room.

The manager believes the adoption rate has dropped below 15%, when it actually has not. The manager will accept more animals into the shelters and will run out of room.

The manager believes the adoption rate is still 15%, when it actually has dropped below 15%. The manager will not accept some animals into the shelters, thinking there will not be enough room, when they could have taken care of those animals.

The manager believes the adoption rate has dropped below 15%, when it actually has not. The manager will not accept more animals into the shelters, when there actually is room to care for those animals.

A group of students is writing a phone message app to provide better word suggestions based on the context of the word and the words the person used in the past. They will earn an A on the project if their teacher is convinced that their program suggests the correct word at least 54% of the time. The current success rate for this app is 54%. The teacher randomly selects 100 words and finds that the students’ program correctly suggests 64 of the words. To determine if this data provide convincing evidence that the proportion of correct words is more than 54%, 150 trials of a simulation are conducted. The results are shown in the dotplot. The teacher is testing the hypotheses: H0: p = 54% and Ha: p > 54%, where p = the true proportion of words the app will correctly predict. Based on the results of the simulation, what is the estimate of the P-value of the test?

0.0133

0.02

0.03

0.0469

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?

The agency concludes that the program increases the recycling rate, when in fact it does not. The program will not be implemented and will save the agency money.

The agency correctly concludes that the program increases the recycling rate. The program will be implemented and will increase water bottle recycling.

The agency concludes that the program increases the recycling rate, when in fact it does not. The program will be implemented at great cost without increasing the rate of water bottle recycling.

The agency concludes that the program does not increase the recycling rate, when in fact it does increase the rate. The agency will not implement the program, and the recycling rate will not improve.

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?

A Type I error would result in failing to reject a false null hypothesis. This means the company would believe the humidity level is at most 3%, when in fact it exceeds 3%.

A Type I error would result in rejecting a true null hypothesis. This means the company would believe the humidity level is more than 3%, when in fact it is not more than 3%.

A Type I error would result in rejecting a true null hypothesis. This means the company would believe the voltage delivered is more than 1,000 volts, when the voltage is actually not more than 1,000 volts.

A Type I error would result in failing to reject a false null hypothesis. This means the company would believe the voltage delivered is no more than 1,000 volts, when the voltage is actually more than 1,000 volts.

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?

The chef believes the complaint rate is less than 8%, when in fact it is not less than 8%. The chef would not use the new recipe, potentially losing customers who need gluten-free menu options.

The chef believes the complaint rate is less than 8%, when in fact it is not less than 8%. The chef continues to use the new recipe but experiences a large number of unsatisfied customers.

The chef believes the complaint rate is not less than 8%, when in fact it is less than 8%. The chef continues to use the new recipe but experiences a large number of unsatisfied customers.

The chef believes the complaint rate is not less than 8%, when in fact it is less than 8%. The chef would not use the new recipe, potentially losing customers who need gluten-free menu options.

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Type I and Type II Errors Answers

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