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The sample mean, often represented by \( \bar{x} \), is a simple statistical metric to find the average of a data set. To calculate the sample mean, add together all the individual ratings given by the sample and then divide the sum by the total number of ratings in the sample. This produces an estimate of the central tendency of the data, giving us an easy way to understand where most data points lie on average. It serves as a crucial element in creating the confidence interval, as it acts as the midpoint of this range. In our example, the sum of all ratings was 343, and with 50 travelers providing these ratings, we obtained a sample mean of \( \bar{x} = \frac{343}{50} = 6.86 \). This means the average rating given by the travelers to Miami International Airport was approximately 6.86 out of 10.

While the sample mean gives us an average, the sample standard deviation (represented by \( s \)) tells us how much the data deviates from this average. It's a measure of the dispersion or variability in the sample ratings. First, calculate the variance by determining the square of differences between each data point and the sample mean, adding those results, and then dividing by one less than the sample size. Afterward, take the square root of this variance to find the standard deviation. In our exercise, the squared differences summed up to 223.68, producing a variance of \( \frac{223.68}{49} \approx 4.564 \). Taking the square root of this variance, we find the sample standard deviation is \( s \approx 2.136 \). A lower value would indicate that the ratings are closer to the mean, while a higher value suggests more variability in traveler ratings.

The t-distribution is pivotal when dealing with small sample sizes, especially when the population standard deviation is unknown. It helps in establishing a range within which we can be confident the true population parameter lies. Unlike the normal distribution, the t-distribution is slightly broader in shape, adjusting for the additional uncertainty introduced by small sample sizes. With 50 samples, we have 49 degrees of freedom (\( n-1 \)). For a 95% confidence interval, the critical t-value obtained from a t-table with 49 degrees of freedom is approximately 2.009. The t-distribution is then applied to the standard error to find the confidence interval around the sample mean. This interval provides a range where the true population mean is expected to fall 95% of the time, offering a balance of accuracy and reliability for our estimates.