Writing Draw a normal curve with a mean of 450 and a standard ...

Question

Writing Draw a normal curve with a mean of 450 and a standard deviation of 50. Describe how you constructed the curve and discuss its features.

Accepted Answer

Below there today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A normal distribution has a mean of 80 and a standard deviation of 10. Which of the following statements is true about the shape and key features of its normal curve? Awesome. So it appears for this particular problem, we're asked to read off our multiple choice answers, and we're asked to determine which of the following statements made in our multiple choice answers is true about the shape and key features of this normal curve based on the information that is provided to us by the prom itself. So now that we know what we're ultimately trying to solve for, let's take a moment to read off our multiple choice answers to see which one might be our final answer. A is the curve is skewed right and about 68% of the data lies between 70 and 90. B is the curve is symmetric, and about 68% of the data lies between 70 and 90. C is the curve is symmetric and about 95% of the data lies between 70 and 90. And finally, D is the curve is skewed left and about 95% of the data lies between 60 and 100. Fantastic. So first off, let us write down all of our known information that is given to us. We're told that the mean, which will denote as mu, is equal to 80, and we're also told that the standard deviation, which will denote as sigma is equal to 10. We are also told that, or rather, we will also note that we know that normal distributions are always symmetric and for normal distributions, 68% of the data lies within plus or minus 1 sigma. And 95% of the data lies between plus or minus 2 sigma, and 99.7% of the data lies within plus or minus 3 sigma. So, for this particular problem, we need to note that mu plus sigma is equal to 80 + 10, which is equal to 90, and mu minus sigma. is going to be equal to. So once again, mu minus sigma is equal to 80 minus 10, which is equal to 70. So that means that undoubtedly, 68% of the data lies between 70 and 90. Awesome. So, with that in mind, we need to consider. Mu + 2 sigma or 2 multiplied by sigma, which is equal to 80 + 2 multiplied by 10. Which is going to be equal to 100. And we also need to consider. minus, so we need to consider mu minus 2 multiplied by sigma, which will be equal to 80 minus 2 multiplied by 10, which is equal to. 60, so we need to note that 95% of the data will lie between 60 and 100. Fantastic. So now let us analyze each of our multiple choice options to see which one is the correct statement. So focusing our attention, so let's go back to look at our multiple choice answers. So let's examine multiple choice answer letter A first. So once again states that the curve is skewed right and about 68% of the data lies between 70 and 90. So the second part of this statement is true, but the first part is incorrect. The curve is not skewed, it is symmetric. So our final answer cannot be A because it means the skew, it's not skewed right, our curve is symmetric. So moving right along here, looking at multiple choice letter B, the curve is symmetric and about 68% of the data lies between 70 and 90. And we need to know with this statement, the curve is symmetric, so that part of the statement is true. And the second part of our statement is also true because our data does indeed lie between 70 and 90. So that means that multiple choice answer letter B is our correct and final answer. But let's look at our other multiple choice answers to see why they are incorrect. So with that in mind, multiple choice answer letter C states the curve is symmetric, and about 95% of the data lies between 70 and 90. So, the first part of this statement is true, but the second part is incorrect because 95% of the data lies between 60 and 100, not 70 and 90, so C is incorrect. So, multiple choice answer letter D states that the curve is skewed left and about 95% of the data lies between 60 and 100. So the data range is correct for the 95% because 95% of the data does indeed lie between 60 and 100, but we once again we established that the curve is symmetric and it is not skewed, so that makes multiple choice answer letter D incorrect. So that's it. We've officially solved for this problem. So our final answer once again, without a doubt, is multiple choice answer letter B, which is the curve is symmetric, and about 68% of the data lies between 70 and 90. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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