In a graduate college, the average length of registration time during a semester is 120 minutes with a standard deviation of 25 minutes. With the introduction of a new registration procedure a random sample of 50 students took an average of 80 minutes with a standard deviation of 12 minutes.

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**Step 1. State the Hypotheses**

We wish to test if the new registration procedure reduces the average registration time. Define the population mean registration time under the new procedure as $$ \mu $$. The hypotheses are:

- $$ H_0: \mu = 120 $$ minutes (the new procedure does not change the time)
- $$ H_1: \mu < 120 $$ minutes (the new procedure reduces the time)

**Step 2. Choose the Appropriate Test Statistic**

Since the historical average registration time and its standard deviation are provided and we assume them to represent the population parameters, we can employ the $$ z $$-test. The test statistic is given by
$$
z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}},
$$
where
- $$ \bar{x} = 80 $$ minutes is the sample mean,
- $$ \mu_0 = 120 $$ minutes is the claimed population mean,
- $$ \sigma = 25 $$ minutes is the population standard deviation,
- $$ n = 50 $$ is the sample size.

**Step 3. Compute the Test Statistic**

First, compute the standard error (SE):
$$
\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{25}{\sqrt{50}}.
$$
Since $$ \sqrt{50} \approx 7.071 $$,
$$
\text{SE} \approx \frac{25}{7.071} \approx 3.536.
$$

Next, calculate the $$ z $$-score:
$$
z = \frac{80 - 120}{3.536} = \frac{-40}{3.536} \approx -11.31.
$$

**Step 4. Determine the p-value and Decision**

For a one-tailed $$ z $$-test, a $$ z $$-score of approximately $$-11.31$$ is extremely low. The corresponding p-value is essentially zero. Since the p-value is far below any common significance level (e.g., $$ \alpha = 0.05 $$ or $$ 0.01 $$), we reject the null hypothesis.

**Step 5. Conclusion**

There is overwhelming statistical evidence that the new registration procedure decreases the registration time.
The new registration procedure significantly reduces the average registration time from 120 minutes to 80 minutes.

In a graduate college, the average length of registration time during a semester is 120 minutes with a standard deviation of 25 minutes. With the introduction of a new registration procedure a random sample of 50 students took an average of 80 minutes with a standard deviation of 12 minutes.

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