Can a large test statistic fail to reject the null if it is in the wrong direction?

No. Direction matters. In a left-tailed test, the rejection region is in the left tail. A positive statistic is on the opposite side of the distribution, so it does not support a less-than alternative.

Example:

• H0: mu >= 6%
• Ha: mu < 6%
• Left-tail critical value at 5%: -1.645
• Test statistic: +2.10

Even though 2.10 has a large absolute value, it is not less than -1.645. Fail to reject the null.

Magnitude is not enough. The statistic must be extreme in the direction specified by the alternative.