The confidence level C and the significance level alpha are linked through the equation
alpha = 1-C
So for instance, if the confidence level is C = 95% = 0.95 then alpha is
alpha = 1-C alpha = 1-0.95 alpha = 0.05
meaning we have a 5% significance level. The larger C gets, the smaller alpha gets and vice versa. It turns out that 0 < C < 1 and also 0 < alpha < 1.
The closer C gets to 1, the alpha value gets closer to 0. The smaller alpha gets, the harder it is to reject the null. Why is that? If we have a fixed p value, say p = 0.02 then we reject the null if alpha > pvalue. But we fail to reject the null when alpha < pvalue. For very small alpha values, we're going to fail to reject H0 no matter how small the pvalue is. The pvalue would have to be really small for H0 to be rejected.
In short, I'm saying that if the confidence level is high, then the chance of rejecting the null hypothesis is low (or rare)
