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import pandas as pd

def add_rolling_mean(df):
    # calculate the 20-day rolling mean for each stock symbol
    rolling_mean = df.groupby(level=0)['sentiment score'].rolling(window='20D').mean()
    
    # create a new column "20d_avg_score" with the rolling mean values
    df['20d_avg_score'] = rolling_mean.reset_index(level=0, drop=True)
    
    return df