A sample tested the claim that heights of men and heights of women have difference​ variances, with s=7.42388 cm for women and 7.14974 cm for men. The sample sizes are n1=144 and n2=156. When using the F test with these​ data, is it correct to reason that there is no need to check for normality because n1>30 and n2>​30?

No. The F test has a requirement that samples be from the normally distributed populations, regardless of how large the samples are.

The F-test simply shows whether the variances that are in the numerator and the denominator are equal. The F-test can be applied on a large sampled population.

One main assumption of the F test is that the populations where the two samples are drawn are normally distributed.

Regarding the question, it's important to note that when using the F test with these​ data, it's not correct to reason that there is no need to check for "normality".

It should be noted that the F test has a requirement that samples are from the normally distributed populations, regardless of how large such samples are.

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nope.   happens to me as well.

-aparri