TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW10.4

Which statement is correct?

In the situation of the two-sample test for proportions, the null hypothesis that the population proportions are equal was not rejected at the 3%-level. (Let the sample sizes be large in the sense that the normal approximation is accurate). Comment on the following statements.

(a) The two observed relative frequencies are equal.

(b) The observed relative frequencies are equal if and only if the absolute frequencies are equal.

(c) The test statistic was larger than the 95%-quantile of the standard normal distribution.

(d) If we had performed a right-sided test, then (c) would be true.

(e) The 99%-confidence interval overlaps zero.

(f) The null hypothesis is correct with probability 97%.

(g) If the null hypothesis holds true, then a type-I error was not made.

Dieses Beispiel ist als solved markiert. Ist dies falsch oder ungenau? Aktualisiere den Lösungsstatus (Details: Vorlage:Beispiel)

Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

a) The two observed relative frequencies are equal.

Not necessarily, it just means that the discrepancy between them is small enough not to reject.

b) The observed relative frequencies are equal if and only if the absolute frequencies are equal.

Only true under the assumption that the sample sizes are equal

c) The test statistic was larger than the $95\%$-quantile of the standard normal distribution.

We only know that it must be outside of $R=(-\infty ,q_{0.015}]\cup [q_{0.985},\infty )$, so it would be smaller than the $98.5\%$ quantile but there is no way of knowing if its larger than the $95\%$ quantile

d) If we had performed a right-sided test, then (c) would be true.

If we would've performed a right sided test, the rejection region $R=[q_{0.97},\infty )$, so no we still wouldn't know anything else other than $z<q_{0.97}$.

e) The $99\%$-confidence interval overlaps zero.

If the $97\%$-confidence interval overlaps zero (as we do not reject the null hypothesis), the $99\%$-confidence interval will surely do so too. A decrease in $\alpha$ only makes the rejection region smaller (we reject less values).

f) The null hypothesis is correct with probability $97\%$.

We cannot make assertions about the correctness of the hypothesis. We just didn't have enough evidence to reject it

g) If the null hypothesis holds true, then a type-I error was not made.

A type-I error happens when the hypothesis is rejected altough it holds true. We didn't reject it so we didn't commit a type-I error.