Testing a Drug A drug manufacturer claims that a drug cures a ...

Question

Testing a Drug A drug manufacturer claims that a drug cures a rare skin disease 75% of the time. The claim is checked by testing the drug on 100 patients. If at least 70 patients are cured, then this claim will be accepted. Use this information in Exercises 31 and 32.

Find the probability that the claim will be rejected, assuming that the manufacturer’s claim is true.

Find the probability that the claim will be rejected, assuming that the manufacturer’s claim is true.

Accepted Answer

Hello, everyone. Let's take a look at this question together. A company produces a new type of LED bulb and claims that 80% of the bulbs last more than 10,000 hours. To verify the claim, a random sample of 120 bulbs is tested. The quality assurance team decides that if fewer than 90 bulbs pass the test, they will reject the company's claim. What is the probability that the claim will be rejected, assuming the company's claim is true? Is it answer choice A 0.0694? Answer choice B, 1.48? Answer choice C 0.694 or answer choice D 0.964. So in order to solve this question, we have to recall how to calculate a probability so that we can calculate the probability that the claim of 80% of the bulbs lasting more than 10,000 hours will be rejected. Assuming that the company's claim is true, and we are using a random sample of 120 bulbs, where the quality assurance team decides that if fewer than 90 bulbs pass the test, they will reject the company's claim. And so the first step in calculating our probability is identifying our given information, which we know provided in the question we are given the following. Which is a claimed success rate of P equals 0.80. Our sample size N is equal to 120. Our rejection threshold is fewer than 90 bulbs last over 10,000 hours. And we know that we need to find the probability that X is less than 90, where we know X is the number of bulbs that pass the test. And since our sample size N is large, we can approximate the binomial distribution with a normal distribution. And so the next step is to find the mean and the standard deviation. Which we know the mean and standard deviation for a binomial distribution. are calculated as the following for the mean we have mu is equal to N multiplied by P, so the mean is 120 multiplied by 0.80, which is equal to 96, and then our standard deviation is equal to the square root of N multiplied by P, multiplied by in parentheses 1 minus P, which plugging in the values into that standard deviation formula, we have a value of approximately 4.38. So our mean is 96 and our standard deviation is approximately 4.38. And now we can use normal approximation with continuity correction, and since we want the probability of X being less than 90, or normal approximation, since we're approximating a discrete distribution with a continuous 1, we apply a continuity correction, so the probability of X being less than 90 is approximately the probability of X being less than or equal to 89.5. And then we convert X equals 89.5 to a Z score, using the formula for the Z score, which is Z is equal to X subtracted by our population mean divided by our population standard deviation, which plugging in the values into the formula, we have 89.5 subtracted by 96, divided by 4.38, so our Z score is approximately 1.48. And now we can find the probability from the standard normal table, which the probability that Z is less than -1.48 is approximately 0.0694. So the probability that the company's claim will be rejected is 0.0694, which looking at our answer choices, we can identify is answer choice A. So answer choice A is the correct answer. And all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

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