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:

Nicotine in Menthol Cigarettes 95% confidence; n = 25, s = 0.24 mg

Nicotine in Menthol Cigarettes 95% confidence; n = 25, s = 0.24 mg

Accepted Answer

Hello. In this video, we're going to be finding the following information given off of the word problem. So we are told that a quality control engineer is interested in the variability of caffeine concentration in a particular brand of espresso. A simple random sample of 30 cups is taken, and the sample standard deviation is found to be S is equal to 0.35 mg per milliliter. Assuming that the population of the concentration is approximately normal, using a 95% confidence level, we want to determine the degrees of freedom, the critical values XL and XR, and the confidence interval up to two decimal places of the population standard deviation. OK, so that was a lot to take in, but let's just go ahead and label what is mainly given to us. So first, we are going to assume that the population of concentrations is approximately normal. And we are told that we have a simple random sample of 30 cups with a sample standard deviation to be S is equal to 0.35 mg per milliliter. So let's just go ahead and write that down. We are going to have N is equal to 30, and S is equal to 0.35. Now, we're also told to use a confidence level, CL of 95%. Now, let's go ahead and approach the first part of this problem, and in fact, let's go ahead and start by first determining the degree of freedom. Now, in order to determine the degree of freedom, the degree of freedom is defined as the sample size minus 1. Now, since we have a sample size of 30, that means our degree of freedom is going to be defined as 30 minus 1, which is equal to 29. So that's going to be the first part of the solution, which is the degree of freedom. But now that we have the degree of freedom, let's go ahead and determine the critical values. Now, on the normal distribution, what we are trying to do with these critical values is we are trying to find the areas of the entails of the normal distribution. That is what X2 L and X2R represent. Now, the first thing we need to do is we need to determine the offset of these tails, which is alpha. In order to find or find the area underneath the tails, alpha is the complement to the confidence level. Since we are given a confidence level of 95%, if we convert that to a decimal, we are given 0.95%, and alpha is going to be 1 minus the 0.95, which is going to be a total of 0.05. Now, in order to find the areas of the entails, that area is going to be defined as alpha divided by 2. That is going to be 0.05 divided by 2, which is going to simplify to be 0.025. So this is going to be the area under the tails to the left and right of the normal distribution. OK. Now let's go ahead and first start by determining our left tail. That is going to be X 2 L. Now we can rewrite this to be X2d. Sub 0.025, which is area with respect to our degree of freedom, which is 29. And by using the appendix for the X square table, we are going to estimate this to be 16.047. That is going to be the value of our left critical point, which is one of the solutions we are looking for. Next, what we are going to do is we are going to determine the area to the right of the rightmost entail, which is going to be X2d sub R. Now, in order to determine the area to the right of this entail, notice that we are looking for the area to the right of a normal distribution. But in order to get the area to the right of the normal distribution, we need to take the entire distribution, which is one, and subtract the value of the area to the left, which is 0.025, and that is going to give us a value of 0.975. So we can rewrite X2 sub R to be X20.975 with respect to our degree of freedom, which is 29. And by using the appendix and the X square table, this is going to give us a value of 45.722, which is the other solution that we are looking for. OK. Now that we have XL and X2R, we can go ahead and use these values to find our confidence interval. Now, the confidence interval is defined as the following. The confidence interval is the value sigma, and it is going to be between two values. The right value is defined as the square root of the degree of freedom multiplied by the standard deviation squared of the normal distribution, and this will be divided by X2d sub L. And on the left hand side, we will take the square root, and this is going to be the degree of freedom multiplied by the standard deviation squared, divided by X 2 R. Now, if we plug in the information, again, the degree of freedom is 29, and we were told that we have a standard deviation of 0.35. We also have the values of X2 subR and X2L, and that is going to allow us to rewrite this confidence interval to be the square root of 29 multiplied by 0.35 squared, divided by the value of X2R, which is 45.722. And sigma, right, the right end point of the sigma confidence interval is going to be the square root of 29 multiplied by 0.35 squared, divided by 16.047. And by using a calculator, we are going to simplify this confidence interval to be 0.28. And 0.47, and these are going to be the values of the confidence interval that we are looking for. So just to recap, by using the given information, we were able to find a degree of freedom that is 29. And by using By taking the degree of freedom, and by finding XL and X sub R, we were able to determine the confidence interval, which tells us that sigma is between 0.28 and 0.47. And with that being said, 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

Chi-Square Distribution

Confidence Interval for Standard Deviation

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