The effect of changing sample sizes on outliers

I know that the size of a sample is inversely proportional to the width of a confidence interval, and that outliers tend to increase the width of the interval as well. So that must mean that increasing the sample size reduces the effect of outliers on a confidence interval, and decreasing the sample size amplifies the effect, correct?

How can I show this using formulae instead of words for, say a confidence interval for a one-sample z-test?

Also as a side note, does changing the sample size change how outliers affect the p-value of a hypothesis test? I'm inclined to say yes, but I'm not sure how to justify that conclusion.

I know that the size of a sample is inversely proportional to the width of a confidence interval, and that outliers tend to increase the width of the interval as well. So that must mean that increasing the sample size reduces the effect of outliers on a confidence interval, and decreasing the sample size amplifies the effect, correct?

How can I show this using formulae instead of words for, say a confidence interval for a one-sample z-test?

Also as a side note, does changing the sample size change how outliers affect the p-value of a hypothesis test? I'm inclined to say yes, but I'm not sure how to justify that conclusion.