Refer to the Bulletin of Marine Science (Apr. 2010) study of...

Refer to the Bulletin of Marine Science (Apr. 2010) study of lobster trap placement, Exercise 7.41 (p. 363). Recall that the variable of interest was the average distance separating traps $-$ called trap spacing-deployed by teams of fishermen fishing for the red spiny lobster in Baja California Sur, Mexico. The trap spacing measurements (in meters) for a sample of 7 teams from the Bahia Tortugas (BT) fishing cooperative are repeated in the table. In addition, trap spacing measurements for 8 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. $$ \begin{array}{lrlllllrl} \hline \text { BT cooperative: } & 93 & 99 & 105 & 94 & 82 & 70 & 86 & \\ \text { PA cooperative: } & 118 & 94 & 106 & 72 & 90 & 66 & 153 & 98 \\ \hline \end{array} $$ 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 a $90 \%$ confidence interval for the target parameter. e. Use the interval, part $\mathbf{d}$, 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 $\mathbf{e}$ to be valid?

Refer to the Bulletin of Marine Science (Apr. 2010) study of lobster trap placement, Exercise 7.41 (p. 363). Recall that the variable of interest was the average distance separating traps $-$ called trap spacing-deployed by teams of fishermen fishing for the red spiny lobster in Baja California Sur, Mexico. The trap spacing measurements (in meters) for a sample of 7 teams from the Bahia Tortugas (BT) fishing cooperative are repeated in the table. In addition, trap spacing measurements for 8 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.
$$
\begin{array}{lrlllllrl}
\hline \text { BT cooperative: } & 93 & 99 & 105 & 94 & 82 & 70 & 86 & \\
\text { PA cooperative: } & 118 & 94 & 106 & 72 & 90 & 66 & 153 & 98 \\
\hline
\end{array}
$$
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 a $90 \%$ confidence interval for the target parameter.
e. Use the interval, part $\mathbf{d}$, 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 $\mathbf{e}$ to be valid?

Key Concepts

Target Parameter

The target parameter represents the true difference between the population means of the two groups being compared. In this context, it is the difference in the average trap spacing between the two fishing cooperatives, which is a central concept in comparative statistics and helps determine if there is a significant difference in behavior between the groups.

Point Estimate

A point estimate is a single value calculated from sample data that serves as the best estimate of a population parameter. For comparing means, the point estimate is typically the difference between the two sample means, which provides an initial insight into the magnitude and direction of the difference between the groups.

Normal versus t-Distribution

When constructing confidence intervals for the difference in means, the normal (z) statistic is not appropriate if the population standard deviations are unknown and the sample sizes are small. Instead, the t-distribution should be used because it adjusts for additional uncertainty in the estimate of the standard error, making it essential for reliable inference under those conditions.

Confidence Interval

A confidence interval provides a range of plausible values for the target population parameter and quantifies the uncertainty associated with the point estimate. In the context of comparing two means, a confidence interval is calculated using the t-distribution to account for the variability in the sample estimates, and it helps in assessing whether the observed difference is statistically significant.

Inference Conditions

For inference regarding two-sample means to be valid, several conditions must be satisfied: the data in each group should be collected from a random or representative sample, observations should be independent both within and between groups, the underlying distributions should be approximately normal (especially when sample sizes are small), and the variances should be similar across the groups if pooling is used. These conditions ensure that the statistical methods applied yield reliable and interpretable results.

Transcript

00:01 Here's the solution to 9 .20.

00:03 We're asked to find a target parameter, and we're looking at lobster trap placement.

00:09 And so we're asked to find the target parameter.

00:11 It even says in the beginning, it says, recall that the variable interest was the average distance separating traps called trap spacing, deployed by teams.

00:19 So the target parameter here is the main trap spacing measurements of the two fishing co -ops.

00:44 All right, now we're supposed to find the point estimate.

00:47 And so to find the point estimate, i'm going to go to the calculator here.

00:51 And if you go to stat, edit, there you can see the data.

00:55 And then i'm going to go to stat and then calc.

00:57 And then i'm going to use two vars stats.

00:59 And i'm just going to find my two.

01:05 I guess they have to be the same, but that's fine.

01:07 So let's go to stat and then tests.

01:11 And then let's do a two sample t test and just use data here.

01:17 So x1 bar is 89.

01:20 We'll say 89 .9 .9.