The government provided further evidence yesterday that the economy had shrugged off a bad winter and the coal strike and has resumed strong, job-creating growth.

The Federal Reserve Board reported that the output of the nation's factories, mines and utilities surged 1.1 percent in April, following a big 1.3 percent rise in March.

Favorable production news was accompanied by a Commerce Department report that builders started new houses at an annual rate of 2.19 million units in April, up 6 percent from March and 15 percent from a year ago. That should trigger production increases in home furnishings and appliances in the months ahead.

Last week, President Carter agreed to reduce both the size and effective late of the tax cut package he proposed last January, in part because inflation is worse than aniticpated and because the economy is growing faster than administration economists had expected.

When he proposed his $25 billion tax cut last January, Carter said it was needed to ensure that the economy continued to grow in late 1978 and in 1979.

Administration economists still believe that the economy runs a strong danger of slowing down at the end of the year, as strong consumer buying slows down. They say that the strong increase in inflation and the difficulties they face with the tax cut in Congress forced the President to scale the tax cut.

Whatever, the prospects for the end of the year, the rest of the spring and summer look strong.

"We got a good, healthy, broad-based gain in April," said the Commmerce Department's deputy chief economist William Cox. "It's the first installment of a good second quarter."

The Federal Reserve said that about one-quarter of the April increase of 1.1 percent was "due to further resumption of coal production" following the end of the three-and-a-half month coal strike in late March.

But, the agency noted, there were increases in output across the board - with the particularly large increase in building of automobilies.

After a bad winter, car sales have surged and the automobile companies are counting on continued sales gains. Auto manufacturers increased the rate at which they are assembling new cars by 6 percent to an annual rate of 9.8 million units in April. That rate is about 2 million cars a year faster than the companies were building them in January.

Automobile output is one of the keys to the total health of the economy. About one in every six American jobs is directly or indirectly dependent upon auto sales.

Aside from automobiles, however, production of other consumer goods - from clothing to staples - rose "only a little after large gains in the two preceding months," the central bank reported.

The Commerce Department's Cox noted that the lack of growth of consumer good production is not disturbing. "Consumers can't buy everything at once," he said.

He pointed to a strong growth in materials, particularly steel, and to another big increase in the production of business equipment. If the economy is to continue to grow late this year and early next, business investment will have to increase sharply.

While the last Commerce Department survey showed that companies only plan to increase their investment this year by 5 1/2 percent in real terms, the continuing growth in production of business equipment may foreshadow an increase in investment about that level this year.

Cox said that part of the strong economic performance in March and April came about "because we slipped on the ice last winter" pushing some of the growth that would have occurred in January and February into March and April.

He said he expects several more months of sustained growth but fears that as consumers overextend themselves in buying toward the end of the year - consumer borrowings relative to their incomes are at a record level - they will cut back. That is source of the administration's fears about slower growth - and fewer jobs created - in 1979.

In another report, the Commerce Department said that businesses added $4.3 billion to their inventories in March, compared with $2.7 billion in February.

Businesses had $100 in inventory for every $142 of sales they made in March, compared with $100 in stock for every $143 of sales they made in February. Manufacturing inventories increased $1.4 billion among manufacturers, $1.5 billion at wholesale and $1.5 billion at retail.