Mean Body Temperature Data Set 5 “Body Temperatures” in Append...

Question

Mean Body Temperature Data Set 5 “Body Temperatures” in Appendix B includes a sample of 106 body temperatures having a mean of 98.20 F and a standard deviation of 0.62 F. Construct a 95% confidence interval estimate of the mean body temperature for the entire population. What does the result suggest about the common belief that 98.6 F is the mean body temperature?

Accepted Answer

Hello. In this video, we are told that a researcher collects a sample of 85 resting heartbeats from heart rates from adults. Finding a mean of 71.4 beats per minute and a standard deviation of 8.7 beats per second, we want to construct a 95% confidence interval for the mean resting heart rate of the population. Then we want to determine if the interval contains the value of 75 beats per minute, often cited as the average. OK, so based off the given information, there are a few things that we can label. First, we are given a sample size. Which we will label with the variable N. And this sample size is given to us as 85. We are given a sample mean. Which we will label as X bar, and this sample mean is 71.4. Next, we are given a sample standard deviation. And this sample standard deviation, which we will label as S is equal to 8.7, and we are told to use a confidence level of 95%, which can also be rewritten as 0.95. OK, so in order to approach this problem, The first thing that we are going to do is we are going to use the idea of a Z score. Since the sample is large enough, we are going to use a Z score for better accuracy. So, the first part of this problem is to get our Z score. Now, the Z score is, well, the first thing we need to do is we first need to determine the area under the tail ends of our normal distribution. So we're going to assume that this data is normally distributed. That area is defined as alpha, and alpha is determined as 1 minus the confidence level. This is going to be 1 minus 0.95, which gives us 0.05. Then, because there are two tail ends to the normal distribution, we want to take this value and divide it by 2. That is going to be 0.05 divided by 2, which gives us 0.025. So, now we want to determine the Z score with an area of 0.025, and by using the Z score table, we get that the critical point is 1.96. This is going to be important for later, so we'll set it to the side for now. OK, now what we want to do is we want to determine the standard error. Now the standard error is going to be defined. As SE, and that is equal to S divided by the square root of N. So in order to get our standard error, we are going to take our sample standard deviation and divide it by the square root of our sample size. That is going to be 8.7 divided by the square root of 85, and with the help of a calculator, this is going to approximate to be 0.944. Next, we need to determine our margin of error. For the margin of error, which is labeled as ME, the margin of error is equal to the Z score we determined earlier, multiplied by our standard error. Now, this is going to give us 1.96 multiplied by the standard error, which is 0.944, and that is going to give us an approximate value of 1.85. And finally, now that we have the margin of error, we can use this to construct the confidence interval. So, the fourth part, which is the confidence interval, is going to be defined, or at least the end points will be defined as X bar plus or minus the margin of error. So in order to determine the endpoints of the confidence interval, we are going to take our sample mean and add and subtract the margin of error to it. That is going to give us the end points of 71.4 minus 1.85 and 71.4 plus 1.85. This is going to simplify to give us the end points of 69.55 and 73.25. Now, in terms of the mean, that means that our mean is going to be between the values of 73.25 and 69.55, and this is going to be the confidence interval that we are looking for. Now, for the second part, we need to determine if 75 beats per minute is within this confidence interval. Since the maximum value of the confidence interval is 73.25, that means that 75 beats per minute is not in the confidence interval. And with that being said, the solution to this problem is going to be B. So, I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Confidence Interval

Standard Deviation

Population Mean vs. Sample Mean

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