Los Angeles Commute Time Listed below are 15 Los Angeles commu...

Question

Los Angeles Commute Time Listed below are 15 Los Angeles commute times (based on a sample from Data Set 31 “Commute Times” in Appendix B). Construct a 99% confidence interval estimate of the population mean. Is the confidence interval a good estimate of the population mean?

Accepted Answer

Hello. In this video, we are told that the following are 15 commute times in minutes for a random sample of Chicago residents. We are given 1035, 2015, 60, 1030, 45, 1055, 2010, 25, 40 and 10. We want to construct a 99% confidence interval for the population mean commute times, and then we want to figure out if the interval is a precise estimate of the mean. OK, so the first thing that we are given is we are given 15 different data points which are acting as a sample of commute times for people in the Chicago Commute Times. We are also told that we want to work with a commute with a confidence level of 99%. So, we have a sample size. Equal to 15, and we have a confidence level equal to 99%, which can be translated to 0.99. So, the first thing we want to do is you want to go ahead and take the mean of the following data table, or the following data, and we want to use those means to construct the square deviation. Let's go ahead and organize our data into a table. So what the following table lists is all of our data values and how we construct the squared deviation. So the first thing that we do is we list out all the given data points. Now, in order to find the mean, which is X bar, we want to take the sum of all of these data points, then divide by the sample size, which is 15. That is going to give us the value of 26.333. So X bar is equal to 26.33, which is our mean. Now, in order to find the square deviation, what we want to do is we want to go ahead and take each of our data points and subtract our mean from them. So, for example, if we take our data point of 10 and subtract 26.33 from it, we get negative 16.333 repeating. And if we square this value, we get 266.77 repeating. So this is how we create the square deviation, and in order to get the square deviation, we take the sum of all the data points within this 3rd table. And the sum of all those data points is going to be 412 or 41, 4,123.33 repeating. So, now that we have the square deviation, we can use this to calculate the sample standard deviation. Now, the sample standard deviation is defined as the following. The sample standard deviation is the square root of the sum of the square deviations. Divided by n minus 1. Now this is going to give us the square root. Of 4123.3333 repeating, divided by 1 less than the sample size, which is 14. And when we plug this into a calculator, this gives us a sample standard deviation of 17.16. Now that we have the sample standard deviation, we want to go ahead and calculate our T score with respect to the 999% confidence in confidence level. So, if we create a T score with respect to the 99% confidence level, and use the T table, this is going to give us a value of 2.977. Now that we've calculated the T score, we can go ahead and use our sample standard deviation and the T score in order to find the sample or the standard error. Now the standard error. Is defined as the following. It is the sample standard deviation divided by the square root of our sample size. So this is going to be 17.16 divided by the square root of 10. And once we plug this into a calculator, we get the value of 4.431. Now that we have the standard error, we can go ahead and calculate the margin of error for our confidence interval. Now, the margin of error is going to be our T score with respect to our confidence interval, multiplied by the standard error. This is going to be 2.977, multiplied by 4.431, and plugging this into a calculator will give us an approximate value of 13.19. And finally, now that we have the margin of error, we can create the confidence interval, or at least calculate the end points. The confidence interval is defined as the sample mean plus or minus the margin of error. This is going to give us 26.33 plus or minus 33.19. And this will simplify to give us a confidence interval of 13.14 minutes on the lower end, and 39.52 minutes on the upper end, and this is going to be the first solution to the problem. Now the second part of the problem asks us. Is the interval a precise estimate of the mean? Now, notice that we have a large difference between the endpoints of the confidence interval. We have 13.14 and we have 39.52. This is a very large gap between the averages of the travel times, and that means that this is not a precise confidence interval. And with that being said, the solution to this problem is going to be C. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

Los Angeles Commute Time Listed below are 15 Los Angeles commute times (based on a sample from Data Set 31 “Commute Times” in Appendix B). Construct a 99% confidence interval estimate of the population mean. Is the confidence interval a good estimate of the population mean?

Key Concepts

Confidence Interval

Sample Mean and Standard Deviation

Normal Distribution and Central Limit Theorem

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