Quality Ratings of Airports. The International Air Transport...

Quality Ratings of Airports. The International Air Transport Association surveys business travelers to develop quality ratings for transatlantic gateway airports. The maximum possible rating is 10 . Suppose a simple random sample of 50 business travelers is selected and each traveler is asked to provide a rating for the Miami International Airport. The ratings obtained from the sample of 50 business travelers follow.$$ \begin{aligned} &\begin{array}{rrrrrrrrrrrrr} 6 & 4 & 6 & 8 & 7 & 7 & 6 & 3 & 3 & 8 & 10 & 4 & 8 \\ 7 & 8 & 7 & 5 & 9 & 5 & 8 & 4 & 3 & 8 & 5 & 5 & 4 \\ 4 & 4 & 8 & 4 & 5 & 6 & 2 & 5 & 9 & 9 & 8 & 4 & 8 \\ 9 & 9 & 5 & 9 & 7 & 8 & 3 & 10 & 8 & 9 & 6 & & \end{array}\\ &\text { Develop a } 95 \% \text { confidence interval estimate of the population mean rating for Miami. } \end{aligned} $$

Quality Ratings of Airports. The International Air Transport Association surveys business travelers to develop quality ratings for transatlantic gateway airports. The maximum possible rating is 10 . Suppose a simple random sample of 50 business travelers is selected and each traveler is asked to provide a rating for the Miami International Airport. The ratings obtained from the sample of 50 business travelers follow.$$
\begin{aligned}
&\begin{array}{rrrrrrrrrrrrr}
6 & 4 & 6 & 8 & 7 & 7 & 6 & 3 & 3 & 8 & 10 & 4 & 8 \\
7 & 8 & 7 & 5 & 9 & 5 & 8 & 4 & 3 & 8 & 5 & 5 & 4 \\
4 & 4 & 8 & 4 & 5 & 6 & 2 & 5 & 9 & 9 & 8 & 4 & 8 \\
9 & 9 & 5 & 9 & 7 & 8 & 3 & 10 & 8 & 9 & 6 & &
\end{array}\\
&\text { Develop a } 95 \% \text { confidence interval estimate of the population mean rating for Miami. }
\end{aligned}
$$

Key Concepts

Central Limit Theorem

The Central Limit Theorem states that, given a sufficiently large sample size, the distribution of the sample mean will approximate a normal distribution regardless of the shape of the population distribution. This theorem justifies the use of normal approximation and t-distribution techniques when constructing confidence intervals for the mean, especially when the sample size is moderate or large.

Margin of Error

The margin of error defines the extent of the uncertainty associated with the sample estimate. It is computed by multiplying the critical value (from the t-distribution for unknown population standard deviation) by the standard error. The margin of error is added to and subtracted from the sample mean to construct the confidence interval, indicating the range within which the true population parameter is likely to lie.

Standard Error

The standard error measures the accuracy with which a sample mean estimates the population mean. It is calculated as the sample standard deviation divided by the square root of the sample size. The standard error provides the basis for constructing confidence intervals, as it quantifies the expected variability of the sample mean from the true population mean due to random sampling.

Simple Random Sampling

Simple random sampling is a fundamental sampling method in which each member of the population has an equal chance of being selected for the sample. This method helps ensure that the sample is representative of the population, minimizes selection bias, and facilitates the validity of statistical inferences made from the sample data.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true value of an unknown population parameter. In inferential statistics, it provides an estimate of the uncertainty surrounding the sample estimate and is associated with a confidence level, such as 95%, which indicates the long-run proportion of intervals that would contain the parameter if the experiment were repeated many times.

t-Distribution

The t-distribution is a probability distribution that is used when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. It accounts for additional uncertainty from using the sample standard deviation, and its shape is similar to the normal distribution but with heavier tails. The degrees of freedom, usually calculated as the sample size minus one, determine the specific form of the t-distribution.

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