To find the number of bacteria in a 4 gram sample, you can use the ratio of the bacteria per gram. Since there are \( 4.1 \times 10^3 \) bacteria in 1 gram, you can multiply this by 4 to find the total for 4 grams:
\[ (4.1 \times 10^3 \text{ bacteria/gram}) \times 4 \text{ grams} = 16.4 \times 10^3 \text{ bacteria} \]
Next, we convert \( 16.4 \times 10^3 \) into proper scientific notation. Since \( 16.4 \) can be expressed as \( 1.64 \times 10^1 \), we can rewrite it as:
\[ 16.4 \times 10^3 = 1.64 \times 10^1 \times 10^3 = 1.64 \times 10^{4} \]
Thus, the amount of bacteria in a 4 gram sample expressed in scientific notation is \( \mathbf{1.64 \times 10^4} \) bacteria.