Q.A firm installs a new machine claiming it raises the average hourly output of a process to at least 250 units, up from the earlier average. A random sample of 36 hours of operation with the new machine shows a mean output of 256 units, with a population standard deviation of 24 units. Test, at the 5% level of significance, whether the new machine's average output is significantly greater than 250 units.

The claim here is specifically directional ('greater than 250'), so this calls for a one-tailed Z-test rather than a two-tailed one, using the one-tailed critical value. …

Step 1 - Hypotheses:

$H_0: \mu = 250$ (no real improvement)

$H_1: \mu > 250$ (one-tailed, since only an increase is being tested)

Step 2 - Level of significance: $\alpha = 5\%$, one-tailed; critical value $= 1.645$.

Step 3 - Test statistic. Given $\bar{x}=256$, $\mu_0=250$, $\sigma=24$, $n=36$:

Check (recompute independently): $\sqrt{36} = 6$ exactly; $24 \div 6 = 4$; $6 \div 4 = 1.5$ — confirmed. …

As with the earlier two-tailed example, this can be cross-checked by converting the critical Z back to output units: the threshold for rejection is 250 + 1.645x4 = 250 + 6.58 = 256.58 units. Since the observed sample mean (256) f …

Using the two-tailed critical value (1.96) instead of the correct one-tailed value (1.645) is the most common error - since the claim here is specifically directional ('greater than'), the one-tailed criti …