Question

Lobster trap placement. Refer to the Bulletin of Marine Science (April 2010) observational study of teams fishing for the red spiny lobster in Baja California Sur, Mexico, Exercise 6.29 (p. 348). Trap-spacing measurements (in meters) for a sample of seven teams of red spiny lobster fishermen are repeated in the table. The researchers want to know how variable the trap-spacing measurements are for the population of red spiny lobster fishermen fishing in Baja California Sur, Mexico. Provide the researchers with an estimate of the target parameter using a 99% confidence interval.

Short answer

From the 99% confidence interval, it can conclude that it is 99% confident that the true variance of trap spacing measurements or red spiny lobster fishermen lies between 43.77546 meters and 1200.99 meters.

Step-by-step solution

Given information

The data represents the trap spacing measurements (in meters) for a sample of seven teams of red spiny lobster fishermen.

Calculating the confidence interval

The 90% confidence interval can be calculated using the formula.

$\frac{\left(n-1\right)s^2}{{\chi }_{\frac{\alpha }{2}}^2}⩽{\sigma }^2⩽\frac{\left(n-1\right)s^2}{{\chi }_{1-\frac{\alpha }{2}}^2}$

From the given data,

The sample mean is calculated using the following formula,

$\overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i$

Therefore,

$\begin{array}{rcl}\overline{x}& =& \left(\frac{93+99+...+86}{7}\right)\\ & =& \frac{629}{7}\\ & =& 89.85\\ & & \end{array}$

Also, sample variance is calculated using the following formula,

${\sigma }^2=\frac{1}{n-1}\sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2$

$$\begin{gathered} =\frac{1}{6}\left({\left(93-89.85\right)}^2+...+{\left(86-89.85\right)}^2\right) \\ =\frac{1}{6}\left({\left(3.15\right)}^2+...+{\left(-3.85\right)}^2\right) \\ =\frac{1}{6}\left(9.9225+...+14.8225\right) \end{gathered}$$

$=135.2569$

Therefore, the standard deviation is

$\begin{array}{rcl}\sigma & =& \sqrt{135.2569}\\ & =& 11.63\\ & & \end{array}$

From the table values, at the 0.01 level of significance and 6 degrees of freedom, the value for ${\chi }_{\frac{\alpha }{2}}^2$is 18.5476, and the value for${\chi }_{\left(1-\frac{\alpha }{2}\right)}^2$ is 0.6757.

Substitute the values to get the required confidence interval.

$$\begin{gathered} \frac{\left(7-1\right){11.63}^2}{18.54758}⩽{\sigma }^2⩽\frac{\left(7-1\right){11.63}^2}{0.67572} \\ 43.7545⩽{\sigma }^2⩽1200.99 \end{gathered}$$

Therefore, the 99% confidence interval ${\sigma }^2$is (43.7546, 1200.99).

From the 99% confidence interval, it can conclude that it is 99% confident that the true variance of trap spacing measurements or red spiny lobster fishermen lies between 43.77546 meters and 1200.99 meters.