A report on the survey will be presented on Tuesday at an event at Gallup headquarters, in Washington, D.C. (This reporter will be part of a panel discussing the findings.) The report and the event are part of a broader campaign by Western Governors, which enrolls about 100,000 students, to prove the value of its teaching model — if not to the department, then at least to policy makers in Congress and other corners of power.

“Do we think the Gallup stuff will shift the department? Probably not,” said Scott Pulsipher, Western Governor’s president. But with Congress now considering a major overhaul of higher-education policy, he said, the value of the mentor model is “an area that policy makers need to consider.”

The stakes remain high for Western Governors. The department’s Office of Inspector General argued in a September audit that the online university had failed to provide “regular and substantive interaction between students and their instructors,” as required by federal law covering distance education, making the institution ineligible to receive federal student aid. The office also recommended that the university return some $700 million in such aid that it had received from July 2014 to June 2016, and possibly more for periods after that.Few higher-education observers expect that the Education Department will actually declare Western Governors ineligible or demand that it repay nearly three-quarters of a billion dollars. After all, the institution has long enjoyed bipartisan support and is widely regarded as a standard-bearer of higher-education innovation, a theme that the education secretary, Betsy DeVos, often sounds.

‘Essential Elements’ of Well-Being

But the department is not ignoring the inspector general’s report either. Western Governors submitted its response to the report in November, and Mr. Pulsipher said his institution and the department have continued to talk about it. “We’ve been actively responding to the department’s review of the OIG audit,” he said.

The Chronicle has asked the university for a copy of its response, but Pulsipher declined, on the advice of its lawyer, he said. The department did not respond to a Chronicle request for a copy, or to a request for comment on the status of the Western Governors review.

Mr. Pulsipher noted that the report had looked only at how the university’s teaching model matched up with regulations, and “made no effort to evaluate the quality.” That is where Western Governors hopes the Gallup survey might become persuasive.

Using Gallup’s traditional measures of what it defines as “well-being” and comparative data on college alumni from the regular surveys it conducts for the Gallup-Purdue Index, the survey covered Western Governors alumni from 2013 to 2017. It found that they were more engaged at work than were alumni of other colleges.

Western Governors alumni were also far more likely to say that their education had been worth the cost, according to the study: 72 percent told Gallup as much, versus 42 percent of alumni of public colleges, 31 percent of alumni of private, nonprofit colleges, and 41 percent of alumni who, like Western Governors students, were older than 25 when they graduated.

The survey also found that a signature element of a Western Governors education — the relationship between students and their program mentors — had been a key factor in their overall well-being. Two-thirds of the alumni strongly agreed that a college mentor had encouraged them to pursue their goals and dreams, a higher proportion than for the other groups of alumni.

Western Governors alumni with strong mentor relationships were also more likely to be seen as thriving in four or five of the “essential elements” of well-being — career, social, financial, physical, and community well-being — that Gallup measured.