Chapter 8: Q 8.78 (page 332)

American Alligators. Multi-sensor data loggers were attached to free-ranging American alligators in a study conducted by Y. Watanabe for the article "Behavior of American Alligators Monitored by Multi-Sensor Data Loggers" (Aquatic Biology, Vol. 18 , pp. 1-8). The mean duration for a sample of $68$ dives was $338.0$ seconds. Assume the population standard deviation is $100$ seconds.
a. Find a $95 %$ confidence interval for the mean duration, $μ$ of an American-alligator dive.
b. Repeal part (a) at a $99 %$ confidence level.
c. Draw a graph similar to Fig. $8.6$ on page 326 to display both confidence intervals.
d. Which confidence interval yields a more accurate estimate of $μ$ ? Explain your answer.

Short Answer

Expert verified

a. The confidence interval for $95 %$ confidence interval of $μ$ is $(314.23, 361.77)$ .

b. The confidence interval for $99 %$ confidence interval of $μ$ is (306.77,369.23).

c. The graph is

d. The $99 %$ confidence level of $μ$ measures more accurate estimate.

Step by step solution

Given Information

Confidence level of $95 %, 99 %$ are given.

$\bar{x} = 338, n = 68 .$

Determining 95% confidence level of μ

It is given that

$\bar{x} = 338, n = 68$

$95 %$ confidence interval is

$\bar{x} \pm z_{\frac{α}{2}} (\frac{σ}{\sqrt{n}}) = 333 \pm 1.96 (\frac{10 π}{\sqrt{68}})$

role="math" localid="1652788209049" $= 338 \pm 23.7685$

role="math" localid="1652788243541" $= (314.23, 361.77)$

Computing confidence interval for 99% for μ

It is given by

$\bar{x} \pm z_{\frac{α}{2}} (\frac{σ}{\sqrt{k}}) = 338 \pm 2575 (\frac{100}{\sqrt{68}})$

$= 338 \pm 312264$

role="math" localid="1652788394394" $= (306.77, 369.23)$

Plotting of graph

For $95 %$ of $μ$

For $99 %$ of $μ$

Explanation of which confidence interval yields a more accurate estimate of μ

$95 %$ confidence interval of $μ$ measure more accurate estimate than the other because of it shorter interval window.

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Short Answer

Step by step solution

Given Information

Determining 95% confidence level of μ

Computing confidence interval for 99% for μ

Plotting of graph

Explanation of which confidence interval yields a more accurate estimate of μ

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