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The appeals process offers the first test of 2010 census results, which found a large-scale population shift to Sun Belt states that tend to lean Republican. In a surprise, blacks in search of jobs increasingly left big cities such as Detroit, Chicago and New York for the suburbs and the South, leading to the first black declines in Michigan and Illinois since statehood.

The challenges won’t affect congressional apportionment and redistricting — revisions to the count don’t affect the redrawing of political boundaries — but they can affect how federal money is handed out.

Population-based federal money goes for programs such as health care, roads and schools. About 60 percent is devoted to Medicaid.

If Houston were to successfully challenge its count as missing 158,475 people based on census estimates released in 2009, Texas could get roughly $948 per person more in Medicaid money, or more than $150 million a year.

There are other effects.

In Detroit, the city’s overall 25 percent decline over the past decade to 713,777 people put the city below the important threshold of 750,000, the level to qualify for some state and federal aid programs, said Mayor Dave Bing, who is challenging the count. One state provision barred Detroit from maintaining its 2.5 percent city income tax rate because of the decline, forcing Michigan lawmakers to pass legislation this month allowing Detroit to keep collecting from taxpayers.

In terms of jobs, “businesses might underinvest in a community because they couldn’t see the true size of the market, say, for a grocery store,” adds Andrew Reamer, a George Washington University public policy professor who wrote a report on the subject for the Brookings Institution, a think tank. “The revenue from federal aid and other sources means cities may be able to borrow less, reduce taxes or spend it on a park or new highway turnoff.”

In New York, Mayor Michael Bloomberg says the city’s 2010 count of 8.2 million missed 200,000 people. He calls it “totally incongruous” that census takers would determine that over half of the 170,000 new housing units added in the city over the last decade were vacant.

Many of the vacancies were in southern Brooklyn and northwestern Queens, which have seen fast growth of working-class immigrants including Chinese, Russians and Arabs. While the census found steady growth in Manhattan and other boroughs, Queens did not grow, and Brooklyn grew by under 2 percent. Some of the homes the city says were missed may have been illegally divided houses with owners reluctant to disclose the number of tenants, who are often undocumented residents.

“The picture the census has drawn of these communities is simply not possible,” said Joseph J. Salvo, director of the population division of the New York City’s Planning Department. “When it says we gained 170,000 housing units but added only 167,000 people, it doesn’t take a demographer to figure out something is wrong. Realtors are calling us and people are asking where the vacant units are, because they want to rent them.”

The average household size in New York City is 2.57 people.

Jersey City, N.J., now in a turnaround after it was afflicted with blight and economic decay decades ago, is paying an outside consultant $25,000 in hopes of proving to the Census Bureau that it has surpassed Newark as New Jersey’s largest city.

Other cities considering challenges based on the costs and potential financial rewards include St. Louis; Atlanta; and the California cities of Santa Ana, San Jose and Long Beach. Faster-growing smaller cities in Texas such as Cibolo, outside San Antonio, and Tyler, near Dallas, already have filed appeals, saying they have evidence that the census used outdated boundary lines or missed pockets of people.

Census Director Robert Groves notes preliminary analyses that show matched, if not improved, performance from 2000 in terms of mail-back rates and reduced duplicates in housing lists. The overall 2010 numbers are also largely consistent with independent U.S. birth and death records, although some initial comparisons suggest the census figure for blacks could have been undercounted by 1.5 percent to 3.8 percent. The government says it is too early to tell whether there was a black undercount without additional analysis, now under way.

The first results from the city challenges are expected this fall, with most appeals taking about six months. A broader assessment of census accuracy is expected sometime next year.