use the given information to find the number of degrees of fre...

Question

use the given information to find the number of degrees of freedom, the critical values X2L and X2R, and the confidence interval estimate of σ. It is reasonable to assume that a simple random sample has been selected from a population with a normal distribution:

Heights of Men 99% confidence; n=153, s=7.10 cm.

Heights of Men 99% confidence; n=153, s=7.10 cm.

Accepted Answer

Hello. In this video, we are going to be finding the following information to the given problem. So we are told that a food scientist wishes to estimate the variability in fiber content of a new high fiber breakfast cereal. A simple random sample of 36 servings yields a sample standard deviation of 120 g. Assuming that the fiber content is normally distributed using a 99% confidence level, we want to determine the degrees of freedom, the critical values Xt subL and the exquit sub-R. And the confidence interval, and we want to estimate this confidence interval up to 2 decimal places for the population standard deviation. OK, this is a lot of information, but let's go ahead and just label some given information. First, we are going to assume that this content is normally distributed. Now, we have a sample size of 36, and this sample has a standard deviation of 120 g. So we are just going to quickly write that down. We have N is equal to 36, and S is equal to 120. We are also told that we are working with a 99% confidence level. So our confidence level CL is equal to 99%. Let's go ahead and start this problem by determining our degree of freedom. Now, in order to find the degree of freedom, The degree of freedom is defined as our sample size N minus 1. Since our sample size is 36, this is going to be 36 minus 1, which is equal to 35. This is going to be the first solution we are working towards in this problem. We have a degree of freedom of 35. Now what we want to do is we want to find the critical values XL. And X squirt sub r. And these critical values are going to be the values of the endpoints of the normal distribution. So, in order to determine this, the first thing we need to do is we need to define the area underneath that curve, which is alpha, or the area underneath the entails of the curve. Alpha is going to be the complement to our confidence level. Now, the confidence level that was given to us is 99%, and if we represent 99% as a decimal, our confidence level is equal to 0.99. Now, since alpha is the complement to the confidence level, alpha is equal to 1 minus 0.99, which is going to give us 0.01. Now, in order to find the areas under the nails, we are going to take this value alpha and divide it by 2. This is going to give us 0.01 divided by 2, which is going to further simplify to be 0.005. Now that we have the areas underneath the entails of the normal distribution, we can use this value, alpha divided by 2, to find XL and XR. So, let's go ahead and start with X2d sub L. Now, X2L again is going to be the area to the left of 0.005. So this is going to be X20.005, with respect to our degree of freedom, which is 35. Now, by using our X square table in the appendix, this is going to have an approximate value of 17.192. This is going to be the first critical point that we are looking for. Now, let's go ahead and determine the second critical point, X2R. Now, first we need to determine the area of R, but the area of R is to the right of that critical point. So, in order to find that area, R is going to be defined as 1 minus alpha, which is 1 minus 0.005. That is going to give us a value of 0.995. So we can represent X2R as X2 0.995 with respect to the degree of freedom, which is 35. And again, by using our X square table in the appendix, that is going to give us a value of 60.275, which is going to be the second critical point that we are looking for. Now that we have the degree of freedom, and we have the two critical points, we can use these pieces of information to find our confidence interval. Now the confidence interval is defined as the following. We have a bounded value of sigma. The right boundary is defined as the square root of the degree of freedom multiplied by the standard deviation squared, and this is going to be divided by X2d sub L. And for the left endpoint of the confidence interval, we have the square root of the degree of freedom multiplied by our standard deviation squared, divided by X2d sub-R. Now, we know that the standard deviation is 1.20. We have our degree of freedom, and we have our critical points. So if we plug these into the confidence interval, we get the square root. Of 35 multiplied by 1.20 squared divided by 60.275. And for the right endpoint, we are going to have the square root of 35 multiplied by 1.20 squared, divided by 17.192. By using a calculator, we can simplify these values to be 0.91 g. And 1.71 g, and this is going to be the confidence interval to the given information. So just to recap, our degree of freedom is 35. Our critical points are 17.192 and 60.275, and the confidence interval is between 0.91 g and 1.71, and these are going to be the solutions to the problem. So I hope this video helps you in understanding on how to find this information, and I will go ahead and see you all in the next video.

Key Concepts

Degrees of Freedom

Critical Values

Confidence Interval for Standard Deviation

Watch next