When one of Mr. Sloane’s lawyers told the judge that his proposed project, in which private school students would play sports with children with disabilities, would bring “the values of inclusion, anti-bullying and diversity directly to the independent school children, who are the focus of this case,” Judge Talwani stopped him, saying that the remarks were “about as tone-deaf as I’ve heard.”

She also expressed irritation at the report from the criminologist, which was submitted by Stephen Semprevivo, another Los Angeles businessman. Mr. Semprevivo has admitted that he paid $400,000 to get his son admitted to Georgetown as a tennis recruit ev en though the son did not play tennis competitively.

Anticipating that other parents scheduled for sentencing in the weeks ahead might follow Mr. Semprevivo’s lead and hire similar consultants, Judge Talwani warned: “I maybe should say to you, before I get nine more of these: I don’t feel I need an expert report from a criminologist to tell me how to rule here, particularly where it’s the same criminologist that’s going to be probably presenting for everybody in L.A.”

Hiring consultants for criminal sentencings is perfectly legal, and former prosecutors said they did not fault defendants for drawing on expert help to try to avoid going to prison. Lawyers for poor defendants also sometimes submit videos or reports from sentencing consultants or mitigation experts during sentencing. But public defenders and court-appointed lawyers have limited budgets for commissioning such materials.

Judge Talwani sentenced Mr. Sloane and Mr. Semprevivo, who were among the first parents in the case to be sentenced, each to four months in prison — less than prosecutors had sought, but more than defense lawyers had argued for. Each man had pleaded guilty to a single count of conspiracy to commit mail fraud and honest services mail fraud; the judge calculated that the four-month sentences fell within federal guidelines.

The actress Felicity Huffman — whose lawyers commissioned and submitted a statistical analysis of sentences for similar defendants — was sentenced to 14 days in prison for conspiring to cheat on her daughter’s SAT exam.

A lawyer for Mr. Semprevivo did not respond to questions about his sentencing presentation. One of Mr. Sloane’s lawyers, Nathan J. Hochman, defended Mr. Sloane’s proposed community service project and said it was an effort not to avoid punishment but to give back to society in a substantive way.