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.

Birth Weights Here are summary statistics for randomly selected weights of newborn girls: n=36, x=3150.0g, s=695.5g (based on Data Set 6 “Births” in Appendix B). Use a confidence level of 95%.

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says a researcher records the systolic blood pressure in millimeters of mercury of 40 randomly selected male patients. The summary statistics are in equal to 40, X bar is equal to 128.5 millimeters of mercury, and S is equal to 18.2 millimeters of mercury. Construct a 95% confidence interval for the population mean. So, the first thing we want to do is determine the number of degrees of freedom that we have. So recall that the degrees of freedom, which we'll label as DF, is going to be equal to N minus 1, wherein here is the number of data points. Now, we're told that we have N as equal to 40 in our problem. That means our degrees of freedom is going to be equal to 40 minus 1. Which is equal to 39. Next, we need to find our critical value. So, to do that, we'll first need to determine our value for alpha, which is going to be the area underneath the tails of our distribution. So recall that alpha is going to be equal to 1 minus our confidence level, which in this case is 95%, but I'll need that as a decimal instead of a percentage, so I'll divide our percentage here by 100, and that will give me 0.95. So that means that alpha is going to be equal to 0.05. I actually just want the area underneath one of the tails, so we'll divide this value by 2, so alpha divided by 2, it's going to be equal to 0.05 divided by 2, which is equal to 0.025. And so, now we can use our degrees of freedom and the area under one of our tails, and a T distribution table in order to find our critical value. So, our critical value, T sub alpha divided by 2. It's going to be equal to, when I look this up in their table. We're going to get a value of 2.023. So now we can use our critical value to calculate our margin of error. So, our margin of error, E, if you recall, is going to be equal to our critical value T sub alpha divided by 2. Multiplied by S divided by the square root of N. Where S here is our standard deviation. Now we're given S in our problem, that was the 18.2, and we were also given N. That means that our margin of error is going to be equal to. 2.023 for a critical value, multiplied by 18.2, or our standard deviation, and that's going to be divided by the square root of 40, where 40 was the number of data points. And when we evaluate this expression, this is going to be approximately equal to 5.822. So here we just rounded 23 decimal places. So now we can calculate the lower bound and upper bound for our confidence interval. Now, for the lower bound, recall that that is going to be X bar. Minus E or X bar here is our mean. And we're given that value as 128.5. So, that means X minus E is going to be equal to 128. 5 minus. 5.822. And this is going to be equal to. 122. 0.7 millimeters of mercury. So here we just rounded to one decimal place. Now, for the upper bound of our confidence interval, we'll have X bar plus E. Which is going to be equal to 128.5. Plus 5.822. And this is going to be equal to 134. 0.3 millimeters of mercury. So that is our upper bound. So, now we can write our confidence interval as an inequality. So we will have 122.7 millimeters of mercury. is less than you. Which is less than 134.3. Millimeter of mercury. So, this here is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Critical Value (t-distribution)

Margin of Error

Confidence Interval

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