WASHINGTON (AP) — Construction of new homes fell 4.7 percent in September, the biggest decline in six months, reflecting weakness in both single-family activity and apartment building. The Commerce Department reports the September result left construction at a seasonally adjusted annual rate of 1.13 million units. It was the sharpest decline since a 7.7 percent fall in March. Homebuilding has been sliding this year, but economists remain optimistic that the low level of unemployment will soon spark a rebound in sales and construction. Even though construction activity has fallen in recent months, home building is 6.1 percent higher than a year ago. Single-family building contracted 4.6 percent in September, while apartment construction was down 5.1 percent.