The data set represents the weights (in pounds) of 10 randomly...

Question

The data set represents the weights (in pounds) of 10 randomly selected black bears from northeast Pennsylvania. Assume the weights are normally distributed. (Source: Pennsylvania Game Commission)

b. Construct a 95% confidence interval for the population mean. Interpret the results.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A wellness researcher is studying the early morning body temperatures in units of degrees Fahrenheit of 10 randomly selected healthy adults. Assume the data follow a normal distribution. The sample data are listed below, and the sample standard deviation is known to be 0.28 °F. 97.5, 98.1. 98.3, 97.8, 98.2, 97.9, 98.4, 98.0, 97.6, 98.1. Using a 95% confidence level, construct a confidence interval for the mean body temperature of healthy adults. Interpret your result. Awesome. So it appears for this particular problem we're asked to use a 95% confidence level to help us construct a confidence interval for the mean body temperature of healthy adults, and then we're asked to interpret our result. So ultimately we're trying to solve for two separate answers combined into one. We're trying to determine the confidence interval for the mean body temperature, and we're asked to interpret that result. So with that in mind, now that we know we're ultimately trying to solve for. Our first step is we need to recall and use the T interval. So we need to use the T interval formula because the population standard deviation is unknown to us. The sample size is small because N is less than 30. And the population is assumed to be normally distributed. So with that in mind, as we should recall, the confidence interval is computed using the following formula, which states that X bar plus or minus T subscript alpha divided by 2. Multiplied by S divided by the square root of N. Fantastic. So now, moving right along here, we need to note that X bar denotes the sample mean T subscript alpha divided by 2 describes the critical T value. S is the sample standard deviation, and N is the sample size. So our second step is we need to compute the sample mean. So, in order to find the sample mean, we need to recall. That in order to find the mean we need to add up all of our data points together and then divide by the number of data points that we have. So that will mean that X bar is equal to 97.5 plus 98.1 + 98.3 plus 97.8 plus 98.2 plus 97.9 plus 98.4 plus 98.0. Plus 97.6 plus 98.1 and then divide by 10 means there's 10 data points. And when we plug that into our calculator, we will find that it's equal to 97.99 °F. Fantastic. So our third step now is we need to identify the T value. So as we should recall, the sample size N is equal to 10. So the degrees of freedom, which is DF for short, is equal to 9 because we need to take 10 minus 1, as we should recall, in order to determine the degrees of freedom. So moving right along here. So for a 95% confidence, that will mean that T subscript 0.025.9. It's equal to 2.6, so it's gonna be 2.262. And we can find this value by using a T distribution table, and there should be a T distribution table in your textbook. Specifically, it would be table 5 if you're using your elementary statistics textbook. Awesome, the 8th edition one. So with that in mind, now that we know that our, once again, our value for T subscript 0.025.9 is equal to 2.262, and we receive that value by using our T distribution table. So our next step now, our 4th step is we need to compute the margin of error, which we can denote the margin of error as E, and the margin of error E is equal to T, subscript alpha divided by 2, multiplied by S divided by the square root of N. Which is equal to when we plug in our known variables, 2.262 multiplied by 0.28 divided by the square root of 10, which will mean when we plug this into our calculator that it'll be approximately equal to 0.20. Fantastic. So now, our 5th step is we need to construct our confidence interval, which is the final answer we're trying to solve for. So let us denote the confidence interval as CI, and the confidence interval is equal to X bar plus or minus the margin of error E. Which is equal to 97.99 plus or minus 0.20, which means that it will be equal to parentheses 97.7998.19. Fantastic. So With that in mind, we need to make our final interpretation, so we are 95% confident that the average early morning body temperature of all healthy adults falls between 97.7. 9 °F and 98.19 °F. So that will mean going back to look at our problem itself, that will mean that our final answer, without a doubt, will have to be the following result. So the result will be once again, to quickly recap what we've learned. It will be parentheses 97.79.98.19 degrees Fahrenheit. We are 95% confident that the population mean lies in this range, and that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

The data set represents the weights (in pounds) of 10 randomly selected black bears from northeast Pennsylvania. Assume the weights are normally distributed. (Source: Pennsylvania Game Commission)

b. Construct a 95% confidence interval for the population mean. Interpret the results.

Key Concepts

Normal Distribution

Confidence Interval

Sample Mean and Standard Deviation

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