Please have a look at zonal statistics and zonal statistics as table tools.

Given the objective, you might want to use Zonal Statistics as Table tool. It will generate a table as output (in contrast to the Zonal Statistics which returns a raster as output). Once you get the table, you may simply join it to your input shapefile to get population for each record.

In Zonal Statistics as Table tool, you would use

1. in_zone_data = Your shapefile (The dataset that defines the zones)

2. zone_field = A unique ID or Name Field for Provinces in shapefile (You would later use this field to join the output table with your shapefile)

3. in_value_raster = Population Grid Raster

4. out_table = The output table that will contain the summary of the values in each zone

5. statistics_type = "SUM" (Because you want to calculate sum of all pixels in each particular province to get cumulative population in that province)

Finally, use Add Join tool to permanently join the output table containing (cumulative average) for each province to the shapefile of the province using zone_field you used in step-2.

Please have a look at zonal statistics and zonal statistics as table tools.

Given the objective, you might want to use Zonal Statistics as Table tool. It will generate a table as output (in contrast to the Zonal Statistics which returns a raster as output). Once you get the table, you may simply join it to your input shapefile to get population for each record.

In Zonal Statistics as Table tool, you would use

1. in_zone_data = Your shapefile (The dataset that defines the zones)

2. zone_field = A unique ID or Name Field for Provinces in shapefile (You would later use this field to join the output table with your shapefile)

3. in_value_raster = Population Grid Raster

4. out_table = The output table that will contain the summary of the values in each zone

5. statistics_type = "SUM" (Because you want to calculate sum of all pixels in each particular province to get cumulative population in that province)

Finally, use Add Join tool to permanently join the output table containing (cumulative population) for each province to the shapefile of the province using zone_field you used in step-2.