band limited white noise is not possible. White noise by definition has a flat power spectral density function.
You can generate a white noise sequence and then filter that sequence to generate a band-limited noise but that noise will not be white.
You do not tell us what the sampling frequency is, which is a critical piece of information to have in designing the filter (you cannot design the filter without it), but I'll assume 2,000 Hz in the following example.
d = fdesign.bandpass('Fst1,Fp1,Fp2,Fst2,Ast1,Ap,Ast2', 150,240,435,550,60,0.5,60,2000);
Hd = design(d);
y = filter(Hd,randn(1000,1));
You can view the filter's magnitude response
I've also given you a minimum-order design with 60-dB of stopband attenuation. If you need less stopband attenuation, you can specify that, or if you wish to specify your filter with a certain order, you can also do that with fdesign.bandpass
