Problem 37 - An Introduction to Mathematical Statistics and Its Applications

Question

Listed below for two consecutive fiscal years are the monthly numbers of passenger boardings at a Florida airport. Use Theorem 14.6.1 to test whether these | Month | Passenger Boardings | Month | Passenger Boardings | | :--- | :---: | :--- | :---: | | July | 41,388 | July | 44,148 | | Aug. | 44,880 | Aug. | 42,038 | | Sept. | 33,556 | Sept. | 35,157 | | Oct. | 34,805 | Oct. | 39,568 | | Nov. | 33,025 | Nov. | 34,185 | | Dec. | 34,873 | Dec. | 37,604 | | Jan. | 31,330 | Jan. | 28,231 | | Feb. | 30,954 | Feb. | 29,109 | | March | 32,402 | March | 38,080 | | April | 38,020 | April | 34,184 | | May | 42,828 | May | 39,842 | | June | 41,204 | June | 46,727 | twenty-four observations can be considered a random sequence, relative to the number of runs up and down. Let \(\alpha=0.05\).

Step-by-Step Solution

Answer

After performing Run's test with critical z-values of ±1.96 (based on \(\alpha=0.05\)) in step 6, if our calculated Z-score lies within this range, it implies that the sequence of passenger boardings could be considered random. Conversely, if it lies outside the range, the sequence is not random. The exact computed Z-score will depend on the number of runs calculated in step 2.

Step 1: Understanding the Pattern

First, merge both years of data and write down the sequence of ups(U) and downs(D). An 'Up' is when the passenger boarding in the current month is higher than the previous month, and a 'Down' is when it is less. If the count is the same, ignore it.

Step 2: Calculating the Number of Runs

Count the total number of runs. A 'run' is a sequence of similar events (either ups or downs) not interrupted by a different event.

Step 3: Calculating the Expected Number of Runs

Using the formula for the expected number of runs, \(E(R) = \frac{{(2n_1n_2)} / {N} +1}\), where \(n_1\) and \(n_2\) are the number of ups and downs, and \(N\) is the total number of observations (ups and downs).

Step 4: Calculating the Standard Deviation

Calculate the standard deviation of the number of runs using the formula \(Var(R) = \frac{{2n_1n_2(2n_1n_2 - N)}}{{N^2(N - 1)}}\) and then square root the variance to get the standard deviation.

Step 5: Calculating the Z-score

Calculate the Z score using the formula \(Z = \frac{{R - E(R)}}{{sd(R)}}\) where \(R\) is the number of runs, \(E(R)\) is the expected number of runs, and \(sd(R)\) is the standard deviation of the number of runs.

Step 6: Testing the Hypothesis

Compare this Z score with the Z score in the Z-table for \(α = 0.05\) (two-tailed test). If the calculated Z score is within the range, then one cannot reject the null hypothesis that the sequence is a random sequence. If it is outside the range, then one rejects the null hypothesis.

Key Concepts

Passenger Boardings Analysis

Evaluating the pattern of passenger boardings at an airport is crucial for forecasting, resource allocation, and operational planning. In our exercise, we analyze the monthly passenger boarding numbers over two consecutive years, seeking to understand whether the sequence of monthly increases and decreases appears random, or if there's an underlying trend or cyclical factor.

By analyzing the data, airport management can make informed decisions about staffing, marketing strategies, and infrastructure development. For the statistical analysis, we use the 'runs test for randomness,' which determines if the sequence of boarding numbers follows a random pattern or not. A non-random pattern may be indicative of a larger trend, and further analysis could unveil seasonal peaks, promotional effects, or other external influences on boarding numbers.

Runs Test for Randomness

The 'runs test for randomness' is a non-parametric test that checks for randomness in a sequence of data points. In practical terms, a 'run' is recognized as a sequence of one or more identical symbols or events. For example, if we're labeling an increase in passenger boardings from one month to the next as an 'up' (U) and a decrease as a 'down' (D), we can get a sequence such as UUDDDUU.

To utilize this test, sort the sequence of data into runs and then use statistical methods to determine if the number of runs is consistent with the number that would be expected in a random sequence. If the result deviates substantially from what's expected, it could imply a non-random pattern in the data, potentially needing further investigation.

Expected Number of Runs

When executing the runs test for randomness, the expected number of runs, denoted by E(R), is a critical component of the analysis. It's calculated based on the number of ups and downs in the data set. The formula for calculating the expected number of runs is \[E(R) = \frac{{(2n_1n_2)} / {N} +1}\] where n₁ and n₂ represent the number of 'ups' and 'downs', and N is the total number of observations that are ups and downs. This calculation gives us a benchmark against which to measure the actual number of runs.

Understanding the expected number of runs helps determine whether the actual run count in the sequence is significantly different from what one would anticipate in a random distribution, providing insight into potential patterns within the data.

Standard Deviation in Statistics

Within the field of statistics, the standard deviation is a measure that indicates the amount of variation or dispersion from the average. It represents how spread out the numbers are in a data set. In the context of the runs test, we calculate the standard deviation of the number of runs to assess the variability in the run sequence.

The formula employed in our exercise is \[Var(R) = \frac{{2n_1n_2(2n_1n_2 - N)}}{{N^2(N - 1)}}\] and taking the square root of this variance gives us the standard deviation. With the calculated standard deviation, we can determine the Z-score, comparing it with critical values from the standard normal distribution to test our hypothesis regarding the randomness of the sequence.