Question

Lobster trap placement. Refer to the Bulletin of MarineScience(April 2010) study of lobster trap placement,Exercise 6.29 (p. 348). Recall that you used a 95% confidenceinterval to estimate the mean trap spacing (in meters)for the population of red spiny lobster fishermen fishing inBaja California Sur, Mexico. How many teams of fishermenwould need to be sampled in order to reduce the width ofthe confidence interval to 5 meters? Use the sample standarddeviation from Exercise 6.29 in your calculation.

Short answer

The 84 teams of fishermen would need to be sampled in order to reduce the width ofthe confidence interval to 5 meters.

Step-by-step solution

Given information

Referring to Exercise 6.29 (p. 348)

Finding the sample standard deviation

No	x
1	93	9.877551
2	99	83.59184
3	105	229.3061
4	94	17.16327
5	82	61.73469
6	70	394.3061
7	86	14.87755
Sum	629	810.8571

Here the values are given

$\sum _{i=1}^7{x}_i=629\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\sum _{i=1}^7{(x_i-\overline{x})}^2=810.8571$

The sample mean is

role="math" localid="1658296119475" $\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum _{i=1}^7{x}_i}}{7}\\ =\frac{629}{7}\\ =89.85714\end{array}$

The sample standard deviation is

$\begin{array}{c}s=\sqrt{\frac{{\displaystyle \sum _{i=1}^7{(x_i-\overline{x})}^2}}{7-1}}\\ =\sqrt{\frac{810.8571}{6}}\\ =11.6251\end{array}$

Finding the sample size

Here the width of the confidence interval is 5

$\begin{array}{c}\text{i.e},2SE=5\\ SE=2.5\end{array}$

Here the standard error is 2.5

The value of the standard deviation is 11.6251

The critical value for a 95% confidence interval is$z_{\alpha /2}=z_{0.05/2}=z_{0.025}=1.96$

$\begin{array}{c}SE=z_{\alpha /2}\frac{s}{\sqrt{n}}\\ n={z^2}_{\alpha /2}\frac{s^2}{S{E}^2}\\ n={1.96}^2\times \frac{{11.6251}^2}{{2.5}^2}\\ n=83.06643\\ n\simeq 84\end{array}$

The required sample size is 84.

Hence 84 teams of fishermen would need to be sampled in order to reduce the width ofthe confidence interval to 5 meters.