Module 14: Background You’ll Need 3

  • Find expected counts based on certain proportions

Flight Frequencies (continued.)

The expected count for each category is the number of trials of the experiment multiplied by the proportion/probability of that particular category.

Delta Airlines would have [latex]92.246962\%[/latex] of its flights arrive on time. Since Delta Airlines had [latex]13,651[/latex] flights arrive in Atlanta in total in March 2021,

[latex]92.246962\% \text{ of } 13,651 = (0.92246962) * 13,651 \approx 12,592.633[/latex]

is the number of flights that we would expect to be on time. This is called the expected count of on-time flights if Delta Airlines’ distribution matched the overall proportions.

Similarly, for Southwest Airlines, we would expect to have [latex](0.92246962)*2,562 \approx 2,363.367[/latex] on-time flights if its distribution matched the overall proportions. Notice that these expected counts do not have to be whole numbers because they are theoretical values.

Notice also that there were [latex]14,956[/latex] total on-time flights for these two airlines in March 2021, so once we knew that Delta Airlines would be expected to have [latex]12,592.633[/latex] on-time flights if its distribution matched the overall proportions, we could have found the expected number of on-time flights for Southwest Airlines by subtracting:

[latex]14,956 - 12,592.633 = 2,363.367[/latex]

We see that we get the same expected count as we did when we used the percentage.

We can also compare the observed and expected counts by calculating the difference between the observed count and the expected count. (So, [latex]\text{observed} - \text{expected}[/latex] for each cell of the table.)

Recall the two-way table that gives the counts for each value of the variable flight status for Delta Airlines and Southwest Airlines arrivals at the Atlanta airport in March 2021.

  On-Time Flights Delayed Flights Canceled Flights Diverted Flights Total
Delta Airlines 12,716 904 23 8 13,651
Southwest Airlines 2,240 299 22 1 2,562

Notice that for Delta Airlines’ on-time flights, the difference is [latex]12,716 - 12,592.633 = 123.367[/latex].

So, Delta Airlines had [latex]123.367[/latex] more on-time flights than would be expected if Delta Airlines’ distribution matched the overall proportions.

  • When the difference between an observed count and the corresponding expected count is positive, it means the expected count was smaller than the observed count, so there were more observed values than expected.
  • When the difference between an observed count and the corresponding expected count is negative, it means the expected count was larger than the observed count, so there were fewer observed values than expected.