In Exercises 5–8, (a) identify the critical value ta/2 used fo...

Question

In Exercises 5–8, (a) identify the critical value ta/2 used for finding the margin of error, (b) find the margin of error, (c) find the confidence interval estimate of u, and (d) write a brief statement that interprets the confidence interval.

Pepsi Weights Here are summary statistics for the weights of Pepsi in randomly selected cans: n=36, x=0.82410 lb, s=0.00570 lb (based on Data Set 37 “Cola Weights and Volumes” in Appendix B). Use a confidence level of 99%.

Accepted Answer

Hello. In this video, we are told that a random sample of cereal boxes yields the following summary statistics. We have a sample size of 25, a sample mean of 18.45 ounces, a sample standard deviation of 0.32 ounce, and using a 95% confidence level, we want to determine the critical values for constructing the margin of error. We want to calculate the margin of error, and we want to compute the confidence interval estimate for the mean. So, in order to start this problem, let's just go ahead and list out the given information. First, we are told that we have a specific sample size, which we will label as the variable n. This sample size is 25. Next, we are given that we have a sample mean, and we are going to label this sample mean as X bar. The sample mean is given to us as 18.45. Next, we are given a sample standard deviation. And we will label the sample standard deviation as S. The sample standard deviation is 0.32. And finally, we are told to use a confidence level. Of 95%. So let's go ahead and approach the first part of this problem, which is that we want to determine the critical values for constructing the margin of error. So, in order to approach Part A, the first thing we want to do is we want to find our degree of freedom. Now, the degree of freedom is defined as N minus 1. So the degree of freedom is going to be our sample size minus 1, which is going to be 25 minus 1 equal to 24. Now what we want to do is we want to go ahead and determine the area under the end points, the critical points. Now that area is going to be going to be defined as alpha, and alpha is going to be 1 minus our confidence level. Now, our confidence level is 95%, which can be rewritten into a decimal as 0.95. So our alpha is going to be 1 minus 0.95, which is equal to 0.05. Now, what we want to do is we want to go ahead and divide this alpha by 2, since we have two tail ends of a normal distribution. So, we have A alpha divided by 2, which is 0.05 divided by 2, and this will simplify to be 0.025. Now what we want to do is we want to use this value and the idea of a T score to determine the critical value. So we are going to determine T sub alpha divided by 2, which is going to be T sub 0.025. And by using the T table in the appendix, we are going to get a value of 2.064. So this is going to be the solution to Part A of the problem. Now what we are going to do is we are going to approach Part B, and Part B asks us to calculate the margin of error. So The margin of error. is defined as the following. The margin of error, ME is equal to our T score T sub alpha divided by 2 multiplied by S divided by the square root of N. So to get the margin of error, we are going to take our T score and multiply it by our sample standard deviation divided by the square root of our sample size. This should be in So, this is going to give us a margin of error, which is 2.064, multiplied by a sample standard deviation, which is 0.32, divided by the square root of 25. And with the help of a calculator, this is going to simplify to be 0.1321. So this is going to be the solution to Part B. Now that we have the T score and the margin of error, we are going to approach Part C, which is computing the confidence interval estimate for the mean. So for part C. The confidence interval estimate is labeled. As X bar minus the margin of error, and X bar plus the margin of error. So in other words, in order to find the endpoints for our confidence interval, we are going to add and subtract our margin of error to the sample mean. That is going to give us 18.45. Minus 2.064 and 18.45 plus 2.064. And this is going to simplify to give us. 18.3179 and 18.5821. And to put this into perspective with the variable new, new is going to be between these values, 18.5. 821 and 18.3179, and this is going to be the confidence interval estimate. So just to recap, We have, we were able to determine the critical value for constructing the margin of error, which is this T score of 2.064. Our margin of error is 0.1321, and the confidence interval for the mean is between the values of 18.3179 and 18.5821. And these are the solutions to the problem. So I hope this video helps you in understanding on how to approach this type of problem, and I will go ahead and see you all in the next video.

Key Concepts

Critical Value (t-distribution)

Margin of Error

Confidence Interval

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