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The sample mean is an essential concept in statistics. It's basically the average of a set of data points collected from a sample, which in this case, is the daily cost of hotel rooms in the Caribbean. To find the sample mean, add up all the prices collected and divide by the number of rooms. In our example, the travel agent found that the average nightly cost for the sample of 52 hotel rooms was $208.35. Calculating the sample mean helps in estimating the average cost in the entire population, which is all hotel rooms in Caribbean countries. It's important because it provides a point estimate that informs us about the general trend within a population. A key point to remember is that the sample mean is an estimate and can fluctuate based on different samples taken from the population.

The population standard deviation is a measure that quantifies how much individual data points in a population deviate from the mean. In simpler terms, it tells us how spread out the prices of all the hotel rooms are around the average price. This is valuable because it gives us an idea of volatility in prices — are they typically very similar, or is there a lot of variation? In our scenario, the population standard deviation for hotel room prices was known to be $47.45. This figure is predetermined for all rooms in Caribbean hotels, not just the sample. This helps in calculating the confidence interval because it portrays the natural dispersion of costs related to the mean cost. Understanding the standard deviation allows better predictions and calculations regarding the average costs.

The Z-score is a statistical measurement that describes a value's relationship to the mean in terms of standard deviations. When we talk about a Z-score in relation to confidence intervals, it represents the number of standard deviations a particular point is from the mean. For our confidence level of 99%, the Z-score used was approximately 2.576. This value is typically derived from standard Z-tables and tells us how extreme an observed result is, assuming the normal distribution. Choosing the correct Z-score is crucial for calculating precise confidence intervals. It fundamentally helps in determining how certain we can be about our estimate of the true mean. A larger Z-score corresponds to greater confidence in our interval estimate, meaning we're more certain our interval contains the true mean.

The standard error (SE) quantifies the variability or the standard deviation of the sample mean estimate of a population mean. In simple terms, it tells us how much the sample mean is expected to fluctuate around the true population mean.The formula for calculating the SE is \[SE = \frac{σ}{\sqrt{n}}\]where \(σ\) is the population standard deviation, and \(n\) is the sample size. For our case, the SE is \[SE = \frac{47.45}{\sqrt{52}}\].Calculating the standard error is a crucial step in forming confidence intervals. It essentially allows us to measure how far we expect the sample mean to be from the population mean. A smaller SE indicates that the sample mean is a more accurate estimate of the population mean, giving us greater confidence in our results.