A soap manufacturing company was distributing a particular brand of a soap through a large number of retail shops. Before a heavy advertisement campaign, the mean sales per week per shop was 140 dozen. After the campaign, a sample of 26 shops was taken and mean sales was found to be 147 dozen with standard deviation 16. Can you consider the advertisement campaign effective ? [Given $t_{25}(0.05) = 2.06$]

This is a one-sample t-test because the population standard deviation is unknown and the sample is small ($n < 30$).

Concept:

• When sigma is unknown and n is small, the sampling distribution of the mean follows Student's t-distribution with $\nu = n - 1$ degrees of freedom. We compare the standardized deviation of the sample mean from the claimed population mean against the critical t value.

Why one-tailed:

• The question asks whether the campaign is 'effective', i.e., whether sales increased, so the alternative is directional ($\mu > 140$) and we use the one-tailed critical value $t_{25}(0.05) = 2.06$.

Decision rule:

• Reject H0 if calculated |t| exceeds the table value. Here $2.23 > 2.06$, so the result is significant and H0 is rejected.

Exam trap:

• Using $\nu = 26$ instead of 25, or forgetting to divide $s$ by $\sqrt{n}$. Both distort the t value.

Real-world link:

• Companies routinely run exactly this A/B style significance test to decide whether marketing spend actually moved sales before scaling a campaign nationally.