Determining Normality. In Exercises 9–12, refer to the indicat...

Question

Determining Normality. In Exercises 9–12, refer to the indicated sample data and determine whether they appear to be from a population with a normal distribution. Assume that this requirement is loose in the sense that the population distribution need not be exactly normal, but it must be a distribution that is roughly bell-shaped.

Taxi Trips The distances (miles) traveled by New York City taxis transporting customers, as listed in Data Set 32 “Taxis” in Appendix B

Taxi Trips The distances (miles) traveled by New York City taxis transporting customers, as listed in Data Set 32 “Taxis” in Appendix B

Accepted Answer

Below there, today we're gonna 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 sample of 20 students scores on a statistics final are 62, 65, 67, 68, 70, 71, 72. 73 74 75 76 77 78 79 80 81 82 83 85 88. Does this sample suggest the scores are from a population that is approximately normal? Awesome. So ultimately it appears our final answer that we're trying to solve for is we're trying to figure out whether or not does this sample suggest that the scores are from a population that is approximately normal. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is yes. The sample suggests an approximately normal distribution, and B is no. The sample does not suggest an approximately normal distribution. Awesome. So, first off, in order to determine whether the sample of 20 students' scores come from a population that is approximately normal, we can take a moment to look at several different features of the data to help us solve for this particular problem. So let us start off our examination of the data features by doing a visual inspection of the data. So we're told from our ordered list, since our list is in numerical order. As you can see, we have 62, 65, 6768, 70, 71 72 73 74, 7576, 77 78 79 80 81 82 8385, and 88. And as we could see, looking at this string of numbers, looking at our ordered list, we can see that it appears to be symmetrical. The values are fairly even and spread around the center. There are no extreme outliers, meaning there are no values that are unusually high or low. We can also see that it is bell-shaped, meaning, as we should recall a note, that the frequency of the scores increases to a center and then decreases. So that's our first step. Our first step is just to look at our order list to see visually, to do a visual inspection of our data, that's our first step. Our second step now, is we need to measure, we need to do a measure of the center and spread, so we can compute some of the basic statistics or the basic stats. So let us determine the mean. So as we should recall, in order to solve for the mean, we need to add all the values together and then divide by 20. So our mean, which we'll say our mean, is equal to 62 + 65 + 67 + 68. Plus 70 + 71 plus 72 plus 73 plus 74 plus 75 plus 76 plus 78. Plus 79 plus 80. Plus 81 plus. 82 + 83 plus 85 plus 88. And then we need to divide everything by 20. So when we do that, we will find that our mean when we round to one decimal place is going to be equal to 75.3, which I'll just call. And let's make sure that we are specific that this is our mean. So our mean is equal to 75.3. So now we need to solve for the median, which, as we should recall and note, which will be the middle of the two scores, which is our 10th and 11th number on our list. So that will mean that our median. is equal to 75 + 76 divided by 2, which will give us an answer of rounded to 1 decial place, 75.5. So that means that the mean and median are close together value-wise, which suggests symmetry. So now our 3rd step is we need to investigate the distribution shape. So, if we were to plot a histogram or a normal quantile, which is a Q-Q plot, we will likely see a relatively smooth unimodal shape. And a Q Q plot with points close to a straight line if the distribution is normal. So that will mean that in conclusion, the data states that it will appear symmetrical. It lacks any outlier values. It will have a roughly bell-shaped spread. And it will also show consistency between the mean and median. So therefore, without a doubt, yes, this sample does suggest the scores come from a population that is approximately normally distributed. So that means our final answer is yes. Hooray, we did it. So this means that our final answer, without a doubt has to be multiple choice answer letter A, which once again is yes, the sample suggests an approximately normal distribution. 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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Key Concepts

Normal Distribution

Central Limit Theorem

Normality Tests

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