91Ó°ÊÓ

A study of a sample of horseshoe crabs on a Florida island (J. Brockmann, Ethology, vol. \(102,1996,\) pp. \(1-21\) ) investigated the factors that were associated with whether female crabs had a male crab mate. Basic statistics, including the five-number summary on weight (kg) for the 111 female crabs who had a male crab nearby and for the 62 female crabs who did not have a male crab nearby, are given in the table. Assume that these horseshoe crabs have the properties of a random sample of all such crabs. $$\begin{array}{lcccccccc}\hline {\text { Summary Statistics for Weights of Horseshoe Crabs }} \\ \hline & n & \text { Mean } & \text { Std. Dev. } & \text { Min } & \text { Q1 } & \text { Med } & \text { Q3 } & \text { Max } \\\\\hline \text { Mate } & 111 & 2.6 & 0.6 & 1.5 & 2.2 & 2.6 & 3.0 & 5.2 \\ \text { No Mate } & 62 & 2.1 & 0.4 & 1.2 & 1.8 & 2.1 & 2.4 & 3.2 \\ \hline\end{array}$$ a. Sketch box plots for the weight distributions of the two groups. Interpret by comparing the groups with respect to shape, center, and variability. b. Estimate the difference between the mean weights of female crabs who have mates and female crabs who do not have mates. c. Find the standard error for the estimate in part b. d. Construct a \(90 \%\) confidence interval for the difference between the population mean weights, and interpret.

Step by step solution

01

Create Box Plots

To create box plots for both groups, use the summary statistics given. Plot the minimum, Q1, median, Q3, and maximum for each group. - For crabs with mates: Min = 1.5, Q1 = 2.2, Median = 2.6, Q3 = 3.0, Max = 5.2. - For crabs without mates: Min = 1.2, Q1 = 1.8, Median = 2.1, Q3 = 2.4, Max = 3.2. Interpret the shape as follows: "Crabs with mates have a slightly higher variability as indicated by a wider box and whiskers. Their data is slightly right-skewed due to a larger max. Crabs without mates have a more symmetric distribution."

02

Calculate Difference in Means

The difference in mean weights is simply the mean of crabs with mates minus the mean of crabs without mates:\[ \text{Difference} = 2.6\, - \,2.1\, = 0.5 \text{ kg} \]

03

Compute Standard Error

The standard error (SE) for the difference in means can be computed using the formula:\[ SE = \sqrt{\frac{(s_1)^2}{n_1} + \frac{(s_2)^2}{n_2}} \]where \(s_1 = 0.6, n_1 = 111\) for crabs with mates, and \(s_2 = 0.4, n_2 = 62\) for crabs without mates. Calculate:\[ SE = \sqrt{\frac{(0.6)^2}{111} + \frac{(0.4)^2}{62}} \approx 0.078 \]

04

Construct Confidence Interval

A 90% confidence interval for the difference in means uses the formula:\[ \text{Difference} \pm t^* \times SE \]With \(df \approx \min(n_1-1, n_2-1) = 61\), for which \(t^* \approx 1.67\) for 90% confidence from the t-distribution table. The interval is calculated as:\[ 0.5 \pm 1.67 \times 0.078 = (0.5 \pm 0.13) = (0.37, 0.63) \]Interpretation: "We are 90% confident that the true difference in population mean weights is between 0.37 and 0.63 kg."

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Box Plot Interpretation

When interpreting box plots, you're looking at a graphical representation of data that shows the distribution's shape, central value, and variability. In our exercise, we're comparing the weight distributions of female horseshoe crabs with mates to those without. A box plot provides a five-number summary consisting of the minimum, first quartile (Q1), median, third quartile (Q3), and maximum.

The width of the box and whiskers gives insights into variability. For crabs with mates, you might notice that the box and whiskers are slightly wider, indicating greater variability. This variability can suggest that there is more spread in the data, meaning the weights are more spread out around the mean. The crabs without mates have a narrower box and whiskers, suggesting less variability and a more consistent weight spread among them.

Shape and skewness can also be interpreted. If one whisker is longer than the other, as in crabs with mates, it indicates skewness. A longer whisker on the right side indicates a right skew (positive skew), suggesting the presence of higher values as outliers. Box plots are especially useful for visually assessing these traits, which help in understanding data dispersion and central tendency without diving into numerical analysis.

Confidence Interval

A confidence interval gives a range of values that is used to estimate a population parameter. It's an important concept in statistical inference, offering an interval that encapsulates the population mean with a certain level of confidence. In our case, we are interested in the difference between the mean weights of female crabs with and without mates.

Suppose you calculate an interval for the difference in means and you obtain a range, such as (0.37, 0.63). At a 90% confidence level, this means that if you were to take 100 different samples and compute a confidence interval for each one, about 90 of these intervals will contain the true mean difference.

Confidence intervals are constructed based on the sample mean and incorporate a margin of error, which is influenced by both the standard error and the chosen confidence level (e.g., 90%, 95%). Higher confidence levels yield wider intervals. This trade-off illustrates the balance between precision and confidence. When interpreting, be sure to understand that it provides a range of plausible values, not a definitive value for the parameter.

Standard Error

Standard error (SE) is a crucial concept when dealing with sample data and making inferences about the population. It gives you an estimate of how much the sample mean would vary if you repeated your experiment multiple times. Essentially, it's a measure of dispersion for the sample mean, similar to how standard deviation measures dispersion for individual data points.

In the context of our exercise, the SE is used to assess the variability of the difference in mean weights between female crabs with mates and those without. You calculate it based on the formula that takes into account the standard deviations and sample sizes of both groups. This gives an insight into how much the mean difference could fluctuate between different samples.

• Lower SE means the sample mean is likely close to the true population mean.
• Higher SE suggests more variability and hence, less reliability in the sample mean.

When constructing confidence intervals, the SE plays a vital role in determining the width of the interval. The smaller the SE, the narrower the interval, and the more precise the estimate.