Question: Refer to the Bulletin of Marine Science (April 2010) study of lobster trap placement, Exercise 6.29 (p. 348). Recall that the variable of interest was the average distance separating traps—called trap-spacing—deployed by teams of fishermen. The trap-spacing measurements (in meters) for a sample of seven teams from the Bahia Tortugas (BT) fishing cooperative are repeated in the table. In addition, trap-spacing measurements for eight teams from the Punta Abreojos (PA) fishing cooperative are listed. For this problem, we are interested in comparing the mean trap-spacing measurements of the two fishing cooperatives.

BT Cooperative	93	99	105	94	82	70	86
PA Cooperative	118	94	106	72	90	66	98

Source: Based on G. G. Chester, “Explaining Catch Variation Among Baja California Lobster Fishers Through Spatial Analysis of Trap-Placement Decisions,” Bulletin of Marine Science, Vol. 86, No. 2, April 2010 (Table 1).

a. Identify the target parameter for this study.b. Compute a point estimate of the target parameter.c. What is the problem with using the normal (z) statistic to find a confidence interval for the target parameter?d. Find aconfidence interval for the target parameter.e. Use the interval, part d, to make a statement about the difference in mean trap-spacing measurements of the two fishing cooperatives.f. What conditions must be satisfied for the inference, part e, to be valid?

Step by step solution

01

(a) Find the target parameter. 

The target parameter will be the population mean and population variance.

02

(b) Compute point estimate. 

For BT Cooperative

$$\begin{gathered} n_1=7 \\ {\overline{x}}_1=89.9 \\ {\sigma }_1=11.6 \\ SEMean=4.4 \end{gathered}$$

For PA Cooperative

$$\begin{gathered} n_2=8 \\ {\overline{x}}_2=99.6 \\ {\sigma }_2=27.4 \\ SEMean=9.7 \end{gathered}$$

The point estimate of the target parameter

$$\begin{gathered} ={\overline{x}}_1-{\overline{x}}_2 \\ =89.9-99.6 \\ =-9.7 \end{gathered}$$

03

(c) State the problem using the z statistic.

The sample size is less than , so we cannot use the normal (z) statistic to find a confidence interval for the target parameter.

04

(d) Find confidence interval.

The degree of freedom will be

$$\begin{gathered} {=n_1+n_2-2}_{} \\ =7+8-2 \\ =13 \end{gathered}$$

From the t-distribution table, the critical value at 0.10 thelevel of significance for 13 degrees of freedom is 1.7709 .

The pooled standard deviation is $s_p=\sqrt{\frac{\left(n_1-1\right){s_1}^2+\left(n_2-1\right){s_2}^2}{n_1+n_2-2}}$

$$\begin{gathered} =\sqrt{\frac{\left(7-1\right){\left(11.6\right)}^2+\left(8-1\right){\left(27.4\right)}^2}{7+8-2}} \\ =\sqrt{\frac{6062.68}{13}} \\ =21.595 \end{gathered}$$

The 90% confidence interval for means difference

$$\begin{gathered} =\left({\overline{x}}_1-{\overline{x}}_2\right)\pm t_{\alpha /2}\times s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}} \\ =\left(89.9-99.6\right)\pm 1.7709\times 21.595\sqrt{\frac{1}{7}+\frac{1}{8}} \\ =-9.7\pm 19.79 \end{gathered}$$

Thus, the confidence interval for means difference is $-29.49\text{to 10.09}$.

05

(e) Make a statement about the difference in mean trap-spacing.

Based on the confidence interval, we can say that the difference between the mean trap spacing measurements of the BT cooperative and the mean trap spacing measurements of the PA cooperative is between $-29.49\text{to 10.09}$.

06

(f) State the conditions that must be satisfied for the inference.

The conditions to be satisfied for the inference in part (e) are:

1. Each sample is taken from simple random sampling.
2. Each sample is independent.
3. Each sample size is small.
4. Each sample is approximately normally distributed.

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