{"id":5493,"date":"2022-09-14T15:52:33","date_gmt":"2022-09-14T15:52:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&p=5493"},"modified":"2022-10-05T12:40:39","modified_gmt":"2022-10-05T12:40:39","slug":"15c-coreq","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/15c-coreq\/","title":{"raw":"15C Coreq","rendered":"15C Coreq"},"content":{"raw":"
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In the next preview assignment and in the next class, you will need to compute relative frequencies and find expected counts based on certain proportions.<\/div>\r\n
Flight Frequencies<\/div>\r\n
In the next in-class activity, we will be comparing distributions of a categorical variable for multiple populations. In particular, we will be looking at whether different airlines have the same distribution of flight status. Our categorical variable will be flight status, and the populations we are comparing are the flights for different airlines. The values of our categorical variable are On-Time, Delayed, Canceled, and Diverted. The following table is a two-way table(also called a contingency table), and it 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.[footnote]U.S.Department of Transportation,Bureau of Transportation Statistics. (n.d.).On-time performance -Reporting operating carrier flight delays at a glance.https:\/\/www.transtats.bts.gov\/HomeDrillChart_Month.asp?5ry_lrn4=FDFD&N44_Qry=E&5ry_Pn44vr4=DDD&5ry_Nv42146=DDD&heY_fryrp6lrn4=FDFE&heY_fryrp6Z106u=F[\/footnote] Notice that each row gives the distribution of flight status for an individual airline.<\/div>\r\n
\r\n\r\n\r\n\r\n\r\n\r\n
<\/td>\r\nOn-Time Flights<\/strong><\/td>\r\nDelayed Flights<\/strong><\/td>\r\nCanceled Flights<\/strong><\/td>\r\nDiverted Flights<\/strong><\/td>\r\nTotal<\/strong><\/td>\r\n<\/tr>\r\n
Delta Airlines<\/strong><\/td>\r\n12,716<\/td>\r\n904<\/td>\r\n23<\/td>\r\n8<\/td>\r\n13,651<\/td>\r\n<\/tr>\r\n
Southwest Airlines<\/strong><\/td>\r\n2,240<\/td>\r\n299<\/td>\r\n22<\/td>\r\n1<\/td>\r\n2,562<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
For Delta Airlines, we can find the relative frequency of on-time flights, or the proportion of on-time flights, by looking at the ratio of on-time flights to the total number of flights: [latex]\\frac{12,716}{13,651}\u22480.9315[\/latex] So, about 93.15% of Delta Airlines\u2019 arriving flights in Atlanta in March 2021 were on time. We can performa similar computation for each value of the variable flight status to obtain the relative frequency distribution of flight status for Delta Airlines in terms of percentages, as shown in the following table.<\/div>\r\n<\/div>\r\n
\r\n
\r\n\r\n\r\n\r\n\r\n
<\/td>\r\nOn-Time Percentage Flights<\/strong><\/td>\r\nDelayed Percentage Flights<\/strong><\/td>\r\nCanceled Percentage Flights<\/strong><\/td>\r\nDiverted Percentage Flights<\/strong><\/td>\r\nTotal<\/strong><\/td>\r\n<\/tr>\r\n
Delta Airlines<\/strong><\/td>\r\n93.15%<\/td>\r\n6.62%<\/td>\r\n0.17%<\/td>\r\n0.06%<\/td>\r\n100%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
\r\n
\r\n

Question 1<\/h3>\r\n1) Complete thefollowingtable for the relative frequency distribution of flight statusfor Southwest Airlines by percentage. Round to the nearest hundredth.\r\n\r\n\r\n\r\n\r\n
<\/td>\r\nOn-Time Percentage Flights<\/strong><\/td>\r\nDelayed Percentage Flights<\/strong><\/td>\r\nCanceled Percentage Flights<\/strong><\/td>\r\nDiverted Percentage Flights<\/strong><\/td>\r\nTotal<\/strong><\/td>\r\n<\/tr>\r\n
Southwest Airlines<\/strong><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n100%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can also consider the total number of flights for each value of the variable flight statusand look at the overall relative frequency for each. For example, there were 12,716+2,240=14,956on-time flights in total for both airlines. The total number of flights overall for both airlines was13,651+2,562=16,213flights. Then,the overall relative frequency for on-time flights was: [latex]\\frac{14,956}{16,213}\u22480.92246962[\/latex] So, about 92.25% of flights were on time for both airlines combined. In this case, we will keep more decimal places to avoid rounding errors in our next computations.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n

Question 2<\/h3>\r\n2) Find the overall relative frequencyexpressed as a percentagefor each value of the variable flight statusand complete the following table.\r\n\r\n\r\n\r\n\r\n
<\/td>\r\nOn-Time Percentage Flights<\/strong><\/td>\r\nDelayed Percentage Flights<\/strong><\/td>\r\nCanceled Percentage Flights<\/strong><\/td>\r\nDiverted Percentage Flights<\/strong><\/td>\r\nTotal<\/strong><\/td>\r\n<\/tr>\r\n
Overall<\/strong><\/td>\r\n92.246962%<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n100%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
Now, let\u2019s imagine that Delta Airlines and Southwest Airlines both had flight status proportions that matched the overall proportions. For example, Delta Airlines would have 92.246962% of its flights arrive on time. Since Delta Airlines had 13,651 flights arrive in Atlanta in total in March 2021, 92.246962% of 13,651=0.92246962\u00d713,651\u224812,592.633is 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\u2019 distribution matched the overall proportions. Similarly, for Southwest Airlines, we would expect to have 0.92246962\u00d72,562\u22482,363.367 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 14,956 total on-time flights for these two airlines in March 2021, so once we knew that Delta Airlines would be expected to have 12,592.633 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:<\/div>\r\n
14,956\u221212,592.633=2,363.367<\/div>\r\n
We see that we get the same expected count as we did when we used the percentage.<\/div>\r\n
\r\n
\r\n

Question 3<\/h3>\r\n3) Complete the following table of expected counts for flights if both airlines had flight status distributions that matched the overall proportions.\r\n\r\n\r\n\r\n\r\n\r\n\r\n
<\/td>\r\nExpected On-Time Flights<\/strong><\/td>\r\nExpected Delayed Flights<\/strong><\/td>\r\nExpected Canceled Flights<\/strong><\/td>\r\nExpected Diverted Flights<\/strong><\/td>\r\nTotal Flights<\/strong><\/td>\r\n<\/tr>\r\n
Delta Airlines<\/strong><\/td>\r\n12,592.633<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n13,651<\/td>\r\n<\/tr>\r\n
Southwest Airlines<\/strong><\/td>\r\n2,363.367<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n2,562<\/td>\r\n<\/tr>\r\n
Total Flights<\/strong><\/td>\r\n14,956<\/td>\r\n1,203<\/td>\r\n45<\/td>\r\n9<\/td>\r\n16,213<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n

Question 4<\/h3>\r\n
4)As we\u2019ve seen in previous in-class activities, we can compare the observed and expected counts by calculating the difference between the observed count and the expected count. (So, observed\u2013expected for each cell of the table.)<\/div>\r\n
Part A: Notice that for Delta Airlines\u2019 on-time flights, the difference is 12,716\u221212,592.633=123.367, so Delta Airlineshad 123.367 more on-time flights than would be expected if DeltaAirlines\u2019distribution matched the overall proportions. Find and interpret the differencefor SouthwestAirlines\u2019delayed flights.<\/div>\r\n
Part B:When the difference between an observed count and thecorrespondingexpected countis negative, itmeans the expected countwas larger than the observed count, so there were fewer observed values than expected. Find and interpret the differencefor SouthwestAirlines\u2019on-time flights.<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"
\n
In the next preview assignment and in the next class, you will need to compute relative frequencies and find expected counts based on certain proportions.<\/div>\n
Flight Frequencies<\/div>\n
In the next in-class activity, we will be comparing distributions of a categorical variable for multiple populations. In particular, we will be looking at whether different airlines have the same distribution of flight status. Our categorical variable will be flight status, and the populations we are comparing are the flights for different airlines. The values of our categorical variable are On-Time, Delayed, Canceled, and Diverted. The following table is a two-way table(also called a contingency table), and it 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.[1]<\/sup><\/a> Notice that each row gives the distribution of flight status for an individual airline.<\/div>\n
\n\n\n\n\n\n
<\/td>\nOn-Time Flights<\/strong><\/td>\nDelayed Flights<\/strong><\/td>\nCanceled Flights<\/strong><\/td>\nDiverted Flights<\/strong><\/td>\nTotal<\/strong><\/td>\n<\/tr>\n
Delta Airlines<\/strong><\/td>\n12,716<\/td>\n904<\/td>\n23<\/td>\n8<\/td>\n13,651<\/td>\n<\/tr>\n
Southwest Airlines<\/strong><\/td>\n2,240<\/td>\n299<\/td>\n22<\/td>\n1<\/td>\n2,562<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
For Delta Airlines, we can find the relative frequency of on-time flights, or the proportion of on-time flights, by looking at the ratio of on-time flights to the total number of flights: [latex]\\frac{12,716}{13,651}\u22480.9315[\/latex] So, about 93.15% of Delta Airlines\u2019 arriving flights in Atlanta in March 2021 were on time. We can performa similar computation for each value of the variable flight status to obtain the relative frequency distribution of flight status for Delta Airlines in terms of percentages, as shown in the following table.<\/div>\n<\/div>\n
\n
\n\n\n\n\n
<\/td>\nOn-Time Percentage Flights<\/strong><\/td>\nDelayed Percentage Flights<\/strong><\/td>\nCanceled Percentage Flights<\/strong><\/td>\nDiverted Percentage Flights<\/strong><\/td>\nTotal<\/strong><\/td>\n<\/tr>\n
Delta Airlines<\/strong><\/td>\n93.15%<\/td>\n6.62%<\/td>\n0.17%<\/td>\n0.06%<\/td>\n100%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
\n
\n

Question 1<\/h3>\n

1) Complete thefollowingtable for the relative frequency distribution of flight statusfor Southwest Airlines by percentage. Round to the nearest hundredth.<\/p>\n\n\n\n\n
<\/td>\nOn-Time Percentage Flights<\/strong><\/td>\nDelayed Percentage Flights<\/strong><\/td>\nCanceled Percentage Flights<\/strong><\/td>\nDiverted Percentage Flights<\/strong><\/td>\nTotal<\/strong><\/td>\n<\/tr>\n
Southwest Airlines<\/strong><\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n100%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

We can also consider the total number of flights for each value of the variable flight statusand look at the overall relative frequency for each. For example, there were 12,716+2,240=14,956on-time flights in total for both airlines. The total number of flights overall for both airlines was13,651+2,562=16,213flights. Then,the overall relative frequency for on-time flights was: [latex]\\frac{14,956}{16,213}\u22480.92246962[\/latex] So, about 92.25% of flights were on time for both airlines combined. In this case, we will keep more decimal places to avoid rounding errors in our next computations.<\/p>\n<\/div>\n<\/div>\n

\n
\n

Question 2<\/h3>\n

2) Find the overall relative frequencyexpressed as a percentagefor each value of the variable flight statusand complete the following table.<\/p>\n\n\n\n\n
<\/td>\nOn-Time Percentage Flights<\/strong><\/td>\nDelayed Percentage Flights<\/strong><\/td>\nCanceled Percentage Flights<\/strong><\/td>\nDiverted Percentage Flights<\/strong><\/td>\nTotal<\/strong><\/td>\n<\/tr>\n
Overall<\/strong><\/td>\n92.246962%<\/td>\n<\/td>\n<\/td>\n<\/td>\n100%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
Now, let\u2019s imagine that Delta Airlines and Southwest Airlines both had flight status proportions that matched the overall proportions. For example, Delta Airlines would have 92.246962% of its flights arrive on time. Since Delta Airlines had 13,651 flights arrive in Atlanta in total in March 2021, 92.246962% of 13,651=0.92246962\u00d713,651\u224812,592.633is 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\u2019 distribution matched the overall proportions. Similarly, for Southwest Airlines, we would expect to have 0.92246962\u00d72,562\u22482,363.367 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 14,956 total on-time flights for these two airlines in March 2021, so once we knew that Delta Airlines would be expected to have 12,592.633 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:<\/div>\n
14,956\u221212,592.633=2,363.367<\/div>\n
We see that we get the same expected count as we did when we used the percentage.<\/div>\n
\n
\n

Question 3<\/h3>\n

3) Complete the following table of expected counts for flights if both airlines had flight status distributions that matched the overall proportions.<\/p>\n\n\n\n\n\n\n
<\/td>\nExpected On-Time Flights<\/strong><\/td>\nExpected Delayed Flights<\/strong><\/td>\nExpected Canceled Flights<\/strong><\/td>\nExpected Diverted Flights<\/strong><\/td>\nTotal Flights<\/strong><\/td>\n<\/tr>\n
Delta Airlines<\/strong><\/td>\n12,592.633<\/td>\n<\/td>\n<\/td>\n<\/td>\n13,651<\/td>\n<\/tr>\n
Southwest Airlines<\/strong><\/td>\n2,363.367<\/td>\n<\/td>\n<\/td>\n<\/td>\n2,562<\/td>\n<\/tr>\n
Total Flights<\/strong><\/td>\n14,956<\/td>\n1,203<\/td>\n45<\/td>\n9<\/td>\n16,213<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
\n
\n

Question 4<\/h3>\n
4)As we\u2019ve seen in previous in-class activities, we can compare the observed and expected counts by calculating the difference between the observed count and the expected count. (So, observed\u2013expected for each cell of the table.)<\/div>\n
Part A: Notice that for Delta Airlines\u2019 on-time flights, the difference is 12,716\u221212,592.633=123.367, so Delta Airlineshad 123.367 more on-time flights than would be expected if DeltaAirlines\u2019distribution matched the overall proportions. Find and interpret the differencefor SouthwestAirlines\u2019delayed flights.<\/div>\n
Part B:When the difference between an observed count and thecorrespondingexpected countis negative, itmeans the expected countwas larger than the observed count, so there were fewer observed values than expected. Find and interpret the differencefor SouthwestAirlines\u2019on-time flights.<\/div>\n<\/div>\n<\/div>\n<\/div>\n
  1. U.S.Department of Transportation,Bureau of Transportation Statistics. (n.d.).On-time performance -Reporting operating carrier flight delays at a glance.https:\/\/www.transtats.bts.gov\/HomeDrillChart_Month.asp?5ry_lrn4=FDFD&N44_Qry=E&5ry_Pn44vr4=DDD&5ry_Nv42146=DDD&heY_fryrp6lrn4=FDFE&heY_fryrp6Z106u=F ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":23592,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5493","chapter","type-chapter","status-publish","hentry"],"part":5479,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5493","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/23592"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5493\/revisions"}],"predecessor-version":[{"id":5620,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5493\/revisions\/5620"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5479"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5493\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5493"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5493"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5493"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5493"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}