Body Temperature Data Set 5 “Body Temperatures” in Appendix B ...

Question

Body Temperature Data Set 5 “Body Temperatures” in Appendix B includes a sample of 106 body temperatures having a mean of and a standard deviation of 0.62F (for day 2 at 12 AM). Construct a 95% confidence interval estimate of the standard deviation of the body temperatures for the entire population.

Accepted Answer

Hello. In this video, we're going to be finding a confidence interval. So we are told that a psychologist measures reaction times for a simple cognitive task in a sample of 50 participants. The sample standard deviation of reaction times is 15 seconds. Assuming reaction times are normally distributed, construct a 95% confidence interval for the true population standard deviation. OK, so let's just go ahead and list some given information. First, we are assuming that this is going to be normally distributed. We are told that this normal distribution has a sample size of 50, and the standard deviation is 0.15 seconds. We are also told that we are working with a 95% confidence interval. So let's just go ahead and list out that information. We have N is equal to 50, S is equal to 0.15, and we have a confidence level equal to 95%, and 95% as a decimal is 0.95. So, in order to find the stand in order to find the confidence interval, we need two pieces of information. We need to find the critical points of the standard deviation, and we need to determine the degree of freedom. So let's go ahead and start with the degree of freedom. Now, in order to find the degree of freedom, the degree of freedom is defined as N minus 1. Now, this is our sample size minus 1, which is going to be 50 minus 1, which is equal to 49. So we have a degree of freedom of 49. Now what we want to do is we want to find the critical points. Of the normal distribution. Now, in order to find these critical points, which we're going to label as XL and X2R, the first thing we need to do is we need to find the areas underneath these critical points. So first we're going to do is we're going to take the complement of the confidence level, which is labeled as alpha. Now, the complement of the of the confidence level is 1 minus the confidence level, which is 0.95. That is going to give us an alpha of 0.05. Now, the area underneath the tail ends of the normal distribution is defined as alpha divided by 2. So that is going to give us 0.05 divided by 2, which simplifies to be 0.025. Now that we have the areas underneath the curves or the underneath the tail ends of the normal distribution, we need to find the critical points to these areas. So, let's go ahead and start with X2d sub L. Now, X subb can be rewritten as X 0.025 with respect to our degree of freedom, which is 49. Now, by using the appendix in the back of the book, which is going to be the table for X2d values and degree of freedoms, that is going to give us a value of 0. Sorry, not 0 point. It is going to be 31. 553. Now what we need to do is we need to find the right critical point, which is X2d sub R. Now, since we are looking for an area to the right of this critical point, we need to find the value of R, which is the complement to the area of the left. That is going to be 1 minus 0.025, which is going to give us a value of 0.975. So, we can represent X2R as X2 0.975 with respect to the degree of freedom, which is 49, and by using the appendix in the back of the book, that will give us the value of 70.222. So these are going to be the two endpoints that we are looking for. Now that we have the critical points and we have the degree of freedom, we have all the information we need to create the confidence interval. Now, the confidence interval is defined as the following. We have our variable sigma, which is between two values. The right value is the square root of the degree of freedom multiplied by the standard deviation squared, divided by X2R. And on the left, uh sorry, X2 L. And on the left hand side. The endpoint is going to be the square root of the degree of freedom multiplied by the standard deviation squared, divided by X2 sub-R. Now, again, the standard deviation was given to us as 0.15. And if we plug in all the information that we found, we can rewrite the confidence interval to be sigma. Which is between the two values on the left hand side, the square root of 49. Multiplied by 0.15 squared, divided by X2 subR, which is 70.222. And on the right hand side, we get the square root of 49. Multiplied by 0.15 squared, divided by 31.553. And with the help of a calculator, we can further reduce this confidence interval to be 0.13. And 0.19. And this is going to be the confidence interval for the study that the scientist is is following, and this is going to be the solution to the problem. So I hope this video helps you in understanding on how to find a confidence interval, and I will go ahead and see you all in the next video.

Key Concepts

Confidence Interval

Standard Deviation

Chi-Square Distribution

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