WASHINGTON — Unemployment rates fell in half of U.S. states last month, a sign that September’s pickup in hiring was felt around the country. The Labor Department says unemployment rates dropped in 25 states, rose in 14 and stayed the same in 11. That’s a modest improvement from August, when unemployment rose in 26 states. Nationwide, employers added 103,000 net jobs in September, nearly double the number created in August. Still, that’s not enough to lower the unemployment rate, which stayed at 9.1 percent for the third straight month. Nevada reported the highest unemployment rate for the 16th straight month. It stayed at 13.4 percent for the second consecutive month. California was next. The rate there fell from 12.1 in August to 11.9 percent. Michigan had the third-highest rate, at 11.1 percent. Layoffs have slowed at a national level in recent months. The number of people applying for unemployment benefits has fallen to a six-month low, according to a four-week average calculated by the government. That has helped calm fears that the economy was sliding into another recession, as have other recent data. Manufacturers in the Philadelphia region grew in October after contracting for two straight months, according to a survey by the Federal Reserve Bank of Philadelphia. In September, consumers boosted their spending on retail goods by the most in seven months. Still, the national unemployment rate has been stuck near 9 percent for more than two years. Employers have added an average of only 72,000 jobs per month in the past five months. That’s far below the 100,000 per month needed to keep up with population growth. And it’s down from an average of 180,000 in the first four months of this year. Americans are pessimistic about the economy. And more than half say President Barack Obama does not inspire confidence about a recovery. A sizable majority — more than 7 in 10 — believe the country is headed in the wrong direction, according to a new Associated Press-GfK poll. And 43 percent describe the nation’s economy as “very poor,” a new high. Among those surveyed, less than 40 percent say Obama’s proposed remedies for high unemployment would increase jobs significantly. Employers pulled back on hiring this spring after seeing less demand from consumers. Higher food and gas prices forced consumers to rein in spending. Consumer spending accounts for 70 percent of economic activity. Job growth is critical to a recovery in the housing market, which many economists say is years away. The number of Americans who bought previously occupied homes fell in September to a seasonally adjusted annual rate of 4.91 million homes, the National Association of Realtors said Thursday. The pace matches last year’s sales figures, which were the worst since 1997.