The government published new statistics yesterday showing a marked step-up in construction spending across the nation in November, bolstering earlier indications that the economy will have made a strong showing for late 1978.

Commerce Department figures showed construction outlays in Novermber up a strong 1.4 percent -- their steepest increase since July. In October, construction spending climbed only 0.3 percent.

The latest round of increases reflected continuing strength in the housing sector of the economy -- particularly in apartment construction, which has been in amini-boom for most of the past several months.

Outlays for homes and apartments grew by a robust 2.4 percent,after declining 0.6 percent in October. At the same time, spending on public consturction projects fell 1.3 percent, following an 0.2 percent dip in October.

The November statistics appeared to underscore other recent economic indicators which showed the economy was not likely to slow down very quickly in late 1978 and early 1979, as had been forecast earlier.

In light of the fourth-quarter performance, most forecasters have pushed of until autumn their predictions for the coming recession. Most see the economy remaining robust right through early spring.

The figures came as, separately, two new industry forceasts yesterday predicted housing whoul slow some this year from 1978's heady pace, but would still turn in a relatiyely good performance, at least by historical standards.

One forecast, by the National Association of Realtors, predicted housing starts this year will drop to about 1.75 million units, a falloff from 1978's 2 million start pace, but still vigorous, especially in the face of high interest rates.

A second, by Bruce Lipke, economist for the Weyerhaeuser Corp., the large Northwest forest products firm, showed the likely number of new starts at 1.6 million. Both are in line with other housing forecasters.

Meanwhile, the Federal Home Loan Bank Board estimated yesterday that if inflation continues at its present pace, the median price of a home in the U.S. will soar to $15,000 within 10 years -- from $50,300 in 1978.

The projections,compiled by Kenneth Biederman, director of the bank's office of economic research, also showed house prices on the West Coast likely to climb even higher -- with a median of $200,000, up from $68,000 now.

The median refers to the price at the midpoint of all home purchases for the year. Currently, almost 13 percent of all single-faimly homes in the U.S. are selling for more than $90,000, up from only 3 percent in 1975.

The West showed the highest median sales prices in 1978. For other regions, the bank board reported a median for $50,000 for homes in the Northeast, $47,000 in the South and $43,000 in the North Central states.

The November statistics on construction spending showed overall outlays at a $212.8 billion annual rate, up from $209.9 billion in October, an overall rise of 1.4 per cent.

After adjustment for inflation, the November rise was 0.7 per cent. By contrast, "real" outlays in October fell 1.1 per cent.