Question 2 • Probability and Statistics with Applications Honors Sem-2-FL-1210300 (2024-2025)
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?
Answer
A
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.
B
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.
C
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.
D
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.
Explanation
A Type I error is rejecting a true null hypothesis. Here, that means the shipment actually contains 75% gold, but the designer concludes it contains less than 75%. A Type II error is failing to reject a false null hypothesis: the shipment contains less than 75% gold, but the designer concludes it contains 75%. These descriptions match choice A.
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