qgis - Problem with spatial reference system / shapefile projection

I am new to using QGIS and having some problems getting some shapefiles to project correctly. I am using publicly available shapefiles for New York City municipal boundaries found here: http://www.nyc.gov/html/dcp/html/bytes/dwndistricts.shtml (ie borough shapefile)

When I download and add the shapefile to my project, the shape is projected in an inaccurate geographic location (as in WAY off).

The extents read as: 835386,112481 : 1144283,277548

And the metadata says:

"General: Storage type of this layer: ESRI Shapefile Source for this layer: /Users/Koren Manning/GIS/nybbwi_12a_av/nybbwi.shp Geometry type of the features in this layer: Polygon The number of features in this layer: 5 Editing capabilities of this layer: Add Features, Delete Features, Change Attribute Values, Add Attributes, Delete Attributes, Create Spatial Index, Fast Access to Features at ID, Change Geometries Extents: In layer spatial reference system units : xMin,yMin 912287.07,116411.37 : xMax,yMax 1067382.51,273617.84 Layer Spatial Reference System: +proj=lcc +lat_1=40.66666666666666 +lat_2=41.03333333333333 +lat_0=40.16666666666666 +lon_0=-74 +x_0=300000 +y_0=0 +ellps=GRS80 +towgs84=0,0,0,0,0,0,0 +units=us-ft +no_defs"

This doesn't make a lot of sense to me but it clearly seems like something is off with the spatial reference system. I tried changing the coordinate reference system to EPSG:4269 - NAD83 which is what other shapefiles I was using were in (from a different source) but this did not change the projection.

Any idea what I am doing wrong?

Any advice is appreciated.

Thank you!