to be considered 18 - karat (18k) gold, a pie...

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 $h_0$: the proportion of metal that is gold is 75%, and $h_a$: 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?\n\ntype i error: the gold shipment truly is made of 75% gold, but the designer concludes that it is made of less than 75% gold.\ntype ii error: the gold shipment truly is made of less than 75% gold, but the designer concludes that it is made of 75% gold.\n\ntype i error: the gold shipment truly is made of 75% gold, but the designer concludes that it is made of less than 75% gold.\ntype 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.\n\ntype i error: the gold shipment truly is made of 75% gold, and the designer concludes that it is made of 75% gold.\ntype ii error: the gold shipment truly is made of less than 75% gold, but the designer concludes that it is made of 75% gold.\n\nmark this and return save and exit next submit

Answer

# Answer: 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. # Brief Explanations: A Type I error occurs when we reject a true null hypothesis ($H_0$). Here, $H_0$ is that the proportion of gold is 75%, so rejecting it when it's true means concluding gold is less than 75% when it's actually 75%. A Type II error is failing to reject a false null hypothesis. So, not rejecting $H_0$ when the gold is truly less than 75% means concluding it's 75% when it's not.