Constructing Confidence Intervals In Exercises 13–24, assume t...

Question

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (b) the population standard deviation σ. Interpret the results.

Drug Concentration The times (in minutes) for the drug concentration to peak when the drug epinephrine is injected into 15 randomly selected patients are listed. Use a 90% level of confidence.

Table displaying drug concentration peak times in minutes for 15 patients, with values ranging from 2.92 to 19.08.

Accepted Answer

Hello everyone. Let's take a look at this question together. The drying times in minutes of a new eco-friendly paint were recorded for a sample of 15 wooden boards. Assume the drying times are normally distributed. The recorded times are as follows, where we have our data values for the drying times in minutes. Construct a 90% confidence interval for the population standard deviation, and also interpret the results. So in order to solve this question, we have to recall how to construct a confidence interval for the population standard deviation, so that we can construct our 90% confidence interval for the drying times in minutes of a new eco-friendly paint, which were recorded for a sample of 15 wooden boards, and we assume. The drying times are normally distributed. And so the first step in solving this problem is organizing our data and computing, which we will first compute the sample mean and the sample variants using the following formula. More specifically, we will use each formula for the sample mean and the sample variants, starting with the Sample mean, which is equal to the sum of our Xi values divided by the sample size N, and using the data that we are provided, we know that the sum of our Xi values is 450 and our sample size N is 15, so 450 divided by 15 is equal to 30. And so our sample mean is equal to 30, which we can then plug in this value into the formula for the sample variants, as we need the sum of our Xi values subtracted by the sample mean squared, which the sum of the Xi subtracted by the sample mean squared is equal to 70. So then we plug in 70 into the sample variance formula, divided by our sample size and subtracted by 1, which gives us 70 divided by 15, subtracted by 1, which is equal to 5. So our sample mean value is 30, and our sample variance value is 5. And now we can determine the degrees of freedom and get our critical values, which we know we have a sample size of 15. So our degrees of freedom, which is calculated as N minus 1, is equal to 14, and we know we have a confidence level of 0.90. So our alpha is equal to 1 minus 0.90, which is equal to 0.10. Therefore, an alpha divided by 2 is equal to 0.05. And then from the chi square table for degrees of freedom equaling 14, we can get our critical values, which our right critical value is equal to 23.685, and our Left critical value is equal to 6.571. So then we use the confidence interval formula for variants, which is given as the following formula, and then we substitute the values into the formula, which gives us 14 multiplied by 5 divided by 23.685, and then 14 multiplied by 5, divided by 6.5. 71. And when simplified, our confidence interval for variance is approximately 2.96 to 10.65. And then we need to take the square root to get the confidence interval of the standard deviation. So the square root of 2.96 to the square root of 10.65 gives us our confidence interval of the standard deviation. As approximately 1.7 to 3.3, and then the interpretation of these results is that we are 90% confident that the true population standard deviation is between 1.7 and 3.3 minutes. And so looking at our answer choices, we can identify that the correct answer choice is answer choice C. And all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Confidence Interval

Population Standard Deviation (σ)

Normal Distribution

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