Q.A packaging machine is set to fill packets with an average of 500 gm of a product, with a known population standard deviation of 40 gm. A random sample of 64 packets is found to have a mean weight of 510 gm. Test, at the 5% level of significance, whether the machine's average fill-weight has changed from 500 gm.

Because the question only asks whether the average has 'changed' (not specifically increased or decreased), this calls for a two-tailed test comparing the sample mean against the claimed 500 gm. …

Step 1 - Hypotheses:

$H_0: \mu = 500$ gm (no change)

$H_1: \mu \neq 500$ gm (two-tailed, since the question only asks if it has 'changed')

Step 2 - Level of significance: $\alpha = 5\%$; critical value for a two-tailed test $= \pm 1.96$.

Step 3 - Test statistic. Given $\bar{x}=510$, $\mu_0=500$, $\sigma=40$, $n=64$:

Check (recompute independently): $\sqrt{64} = 8$ exactly; $40 \div 8 = 5$; $10 \div 5 = 2$ — confirmed by direct recalculation. …

An equivalent approach is to convert the critical Z of 1.96 back into fill-weight units: the 'acceptance region' for x-bar is 500 +/- 1.96x5 = 500 +/- 9.8, i.e. (490.2, 509.8) gm. Since the observed x-bar = 510 …

Using a one-tailed critical value (1.645) here would be wrong, since the question asks only whether the mean has 'changed', not whether it has specifically increased - that requires the two-tailed critical value 1.96. Also, forgetting …