A nutritionist believes that the mean daily calorie intake is higher for athletes than for non-athletes. In a sample of $16$ athletes, the mean intake is $3,150$ calories with a standard deviation of $410$. In a sample of $20$ non-athletes, the mean intake is $2,900$ calories with a standard deviation of $350$. At $\alpha =0.05$, is there enough evidence to support the nutritionist's belief? Assume population variances are not equal.