Approximating Binomial Probabilities In Exercises 19–26, deter...

Question

Approximating Binomial Probabilities In Exercises 19–26, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Identify any unusual events. Explain.

Athletes on Social Issues In a survey of college athletes, 84% said they are willing to speak up and be more active in social issues. You randomly select 25 college athletes. Find the probability that the number who are willing to speak up and be more active in social issues is (b) less than 23

Accepted Answer

All right, hello, everyone. So this question says, a factory produces light bulbs, and 85% of them has quality control. If a box contains 75 bulbs, what is the probability that fewer than 68 bulbs pass quality control? Use the normal approximation if appropriate, and state whether this is an unusual event. So for this question, let's say that X is the number of bulbs out of 75 that pass quality control. And X happens to follow a binomial distribution. Now, specifically, N is equal to 75, that's the sample size, and P is equal to 0.85. That's the probability that the bulb selected passes quality control. So first, we have to check if the normal approximation makes sense for this case. So to do that, we have to check both NP and N multiplied by 1 subtracted by P. So NP is equal to 75 multiplied by 0.85. And this equals 63.75. On the other hand, we have 75 multiplied by 1 subtracted by 0.85. And this equals 11.25. Now both of these values are greater than 5. Which means that the normal approximation is indeed appropriate. So from there We move on to finding the mean and standard deviation. So the mean or mu recalls equal to NP. So we already know that that's Really, that's 63.75 or 75. And so the standard deviation sigma. Is equal to the square root of NP. Multiplied by 1 subtracted by P. And so that's equal to. 75 Multiplied by 0.85. Multiplied by 1 subtracted by 0.85. So this expression simplifies to the square root of 9.5625. So now What we're solving for. Is PO. X less than 68. But we do have to apply the continuity correction. So for the continuity correction, we need to subtract 0.5 from 68. So Ultimately, our expression comes out to P of. X Less than 67.5. And so the Z score that we have to calculate is based on 67.5. So Z Is equal to 67.5. Subtracted by 63.75. And divided by sigma. Which is the square root of 9.5625? Ultimately, Z is equal to approximately 1.2127. And Now, the task is to find the corresponding probability using the standard normal distribution tape. But notice how our calculated Z score is actually between two values on the standard normal table. Specifically, it's between 1.21. And 1.22. So from the standard normal distribution table, The probability of Z less than 1.21. Is 0.8869 Whereas the P of Z less than 1.22 is 0.8888. So we need to interpolate. P of Z less than 1.2127. is equal to 1.2127. Subtracted by 1.21. Divided by 1.22 subtracted by 1.21. Now this term is then multiplied by 0.8888. Subtracted by 0.8869. And then you add 0.8869. And so this equals approximately 0.8874. So, the probability that fewer than 68 bulbs pass quality control is 0.8874. Which now leaves you with interpretation. Right, is this event unusual? Well, recall that in order for an event to be considered unusual, the probability has to be. Less than oops. 0.05, but 0.8874 is instead greater than 0.05. Which means That this is not An unusual event. And there you have it. So now that completes your final answer. And so with that being said, thank you so very much for watching, and I hope you found this helpful.

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Key Concepts

Binomial Distribution

Normal Approximation to the Binomial

Probability Calculation

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