Question 20 A marine biologist would like to estimate the mean lifespan of all Florida manatees. The mean lifespan of a random sample of 49 Florida manatees was found to be 51.5 years. The standard deviation of the lifespans of all Florida Manatees is known to be 6.6 years. Using a 99.9\% confidence level, determine a confidence interval for the mean lifespan of all Florida Manatees. Report the confidence interval using interval notation. Round solutions to two decimal places, if necessary. A $99.9 \%$ confidence interval is given by Suppose the population standard deviation was unknown, but the marine biologist found the standard deviation of the lifespan of the sample of 49 Florida manatees was 7.9 years (along with the sample mean of 51.5 years). Determine a $99.9 \%$ confidence interval for the mean lifespan of all Florida manatees using this new information A $99.9 \%$ confidence interval is given by Question Help: $\square$ Video Submit Alt Parts $75^{\circ} \mathrm{F}$ Mostly sunny Search

Step 1 ：The formula for a confidence interval is: \(\text{mean} \pm \text{z-score} \times \left(\frac{\text{standard deviation}}{\sqrt{\text{sample size}}}\right)\)

Step 2 ：First, we need to find the z-score for a 99.9% confidence level. The z-score for a 99.9% confidence level is 3.29.

Step 3 ：When the population standard deviation is known, the mean lifespan is 51.5 years, the standard deviation is 6.6 years, and the sample size is 49.

Step 4 ：So, the confidence interval is: \(51.5 \pm 3.29 \times \left(\frac{6.6}{\sqrt{49}}\right)\)

Step 5 ：Simplify the expression to: \(51.5 \pm 3.29 \times \left(\frac{6.6}{7}\right)\)

Step 6 ：Further simplify to: \(51.5 \pm 3.29 \times 0.94\)

Step 7 ：Finally, we get: \(51.5 \pm 3.09\)

Step 8 ：\(\boxed{\text{So, the 99.9% confidence interval for the mean lifespan of all Florida manatees is (48.41, 54.59) years.}}\)

Step 9 ：When the population standard deviation is unknown, the mean lifespan is 51.5 years, the standard deviation is 7.9 years, and the sample size is 49.

Step 10 ：So, the confidence interval is: \(51.5 \pm 3.29 \times \left(\frac{7.9}{\sqrt{49}}\right)\)

Step 11 ：Simplify the expression to: \(51.5 \pm 3.29 \times \left(\frac{7.9}{7}\right)\)

Step 12 ：Further simplify to: \(51.5 \pm 3.29 \times 1.13\)

Step 13 ：Finally, we get: \(51.5 \pm 3.72\)

Step 14 ：\(\boxed{\text{So, the 99.9% confidence interval for the mean lifespan of all Florida manatees is (47.78, 55.22) years.}}\)