Construction About 63% of the residents in a town are in favor...

Question

Construction About 63% of the residents in a town are in favor of building a new high school. One hundred five residents are randomly selected. What is the probability that the sample proportion in favor of building a new school is less than 55%? Interpret your result.

Accepted Answer

All right, hello, everyone. So this question says a recent survey found that 72% of adults in a city support expanding the local park. If 120 adults are randomly chosen, what is the probability that the sample proportion? Supporting the expansion is less than 65%. And here we've got 2 different, excuse me, 4 different answer choices labeled A through D. So let's point out the information that we know. First and foremost, we have the population proportion, that's P. And we know that's equal to 0.72. We also know that N, our sample size, is 120 adults. So let's allow for Phat to be the sample proportion. And so what we're looking for in this question is P or probability of Phat being less than 0.65. So now, let's check if the normal approximation is appropriate here. And to do this, we have to check both NP. And N multiplied by 1 subtracted by P. So, NP is equal to 120, multiplied by 0.72. Which equals 86.4. Now n multiplied by 1 subtracted by P. is equal to 120 multiplied by One is subtracted by 0.72, which gives you 0.28. And this is equal to 33.6. So both of these numbers. are greater than 10. And because of this, we can assume that the sampling distribution of Phat is approximately normal. So let's proceed. And the next step now is to find the mean and then the standard deviation of the sampling distribution. So Starting off with the mean mew of pea hat. In this case, would just be equal to P, which is 0.72. So now the standard deviation, that's sigma of Phat. Is equal to the square root of. P multiplied by one subtracted by P. Divided by N So that's equal to. 0.72 Multiplied by 0.28. And divided by 120. After rounding, this gives you approximately 0.041. So now we have to find our Z score. The Z score that corresponds to 0.65. is equal to 0.65, subtracted by mu, that's 0.72. And divided by sigma, that's 0.041. So now this gives you approximately -1.707. And after referencing the standard normal table, you'll notice that our calculated Z score is actually in between two listed on the. Table. Specifically, it's between. -1.70. And -1.71. And so Their corresponding probabilities. are equal to 0.0446. And 0.0437. Keep in mind that this is the probability of Z being less than either of these two values. So To find this e score that we're looking for, we are going to have to use linear interpolation. So Scrolling down here to open up some more space. The probability ultimately, of Z being less than -1.707. Is approximately equal to P. Sub 1 added to 0.7, multiplied by delta P. So let's clarify what this means, right? Piece of one in this case, corresponds to the probability of Z being less than -1.70, since that's the first. Value listed here And delta P is the difference between both probabilities from the standard normal table. So Delta P is equal to 0.0437. Subtracted by 0.0446. Which equals approximately 0.0009. So going back to our probability. And substituting the information. That's going to be equal to. 0.0446 Added to 0.7. Multiplied by 0.009. And this Equals 0.0440. And there you have it. So now, scrolling up here to see the answer choices. 0.0440 corresponds to option D in the multiple choice. And so with that being said, thank you so very much for watching, and I hope you found this helpful.

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