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A confidence interval is a range of values used to estimate a population parameter. In the context of this study, we're looking for a 95% confidence interval for the mean carapace length of lobsters. This implies that we are 95% confident that the true mean lies within this interval. The idea is to give a range instead of a precise number as our estimate based on the sample data.
This interval is calculated using the mean of the sample, the standard deviation, and a critical value from the t-distribution, since the sample size is relatively small (less than 30), making the t-distribution more appropriate than the normal distribution.
The general formula for a confidence interval is:

• CI = \(\mu \pm t_{critical} * \frac{s}{\sqrt{n}}\)

This encompasses:

• the sample mean \(\mu\)
• the t-critical value that accounts for a particular confidence level
• the estimated standard error calculated from the sample standard deviation \(s\)
• the square root of the sample size \(n\)

This systematic approach gives us a statistically grounded range in which the true mean likely lies.

Mean calculation is the first step in finding the confidence interval. The mean, symbolized as \(\mu\), is the average of the data set. It's calculated by summing all the data points and dividing by the number of points.
In the lobster study, we sum up the length of carapaces from 10 lobsters and then divide by 10:
\(\mu = \frac{78 + 66 + 65 + 63 + 60 + 60 + 58 + 56 + 52 + 50}{10} = 60.8\) mm
This means, on average, the carapace lengths were around 60.8 millimeters.
The mean is a critical value as it serves as the center of our confidence interval. It provides a basic understanding of where most of our data lies and is used alongside the standard deviation to understand data variability.

The standard deviation \(s\) is a measure of the amount of variation within a set of data. A low standard deviation indicates that the data points are close to the mean, whereas a high standard deviation indicates that the data points are spread out over a wider range.
In our study, the standard deviation is calculated using:
\(s = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\mu)^2}{n-1}}\)
This involves subtracting the mean from each data point, squaring the result, summing all these squared differences, then dividing by \(n-1\) which accounts for Bessel's correction in small samples. The square root of this quotient gives us the standard deviation.
Here, the standard deviation came out to 9.07 mm, indicating quite some variability among the lobster carapace lengths. This variability impacts the width of our confidence interval; high variability can lead to a wider interval, suggesting less precision in our estimation of the mean.

Degrees of freedom (df) refer to the number of values in a calculation that are free to vary. In the context of confidence intervals, it affects the precision with which we can estimate population parameters.
For a sample mean, degrees of freedom are calculated as \(n - 1\), where \(n\) is the sample size. This reduction by 1 is important because we estimate one parameter, the mean, from the data set.
In our example with 10 lobsters, the degrees of freedom are \(10 - 1 = 9\).
Degrees of freedom are crucial when using the t-distribution to account for variability. With fewer degrees of freedom, the t-distribution becomes wider, which typically results in wider confidence intervals. As the sample size grows and with it the degrees of freedom, the t-distribution approaches the normal distribution, offering more precise estimations.