Question 1168567
Let's break down this hypothesis test step-by-step.

**1. Statistical Hypotheses**

* **Null Hypothesis (H₀):** The mean time taken by the students is 20 minutes (μ = 20).
* **Alternative Hypothesis (H₁):** The mean time taken by the students is different from 20 minutes (μ ≠ 20).

This is a two-tailed test because we are testing if the time is *different* from 20 minutes, not specifically less than or greater than.

**2. Reject the Null Hypothesis**

* Significance level (α) = 0.05
* Sample size (n) = 48
* Degrees of freedom (df) = n - 1 = 48 - 1 = 47

Since the sample size is large (n > 30), we can use the t-distribution or approximate it with the z-distribution. However, it's safer to use the t-distribution with df = 47.

* Using a t-table or calculator:
    * t(0.025, 47) ≈ ±2.012 (This is the critical t-value for a two-tailed test with α = 0.05 and df = 47).

* Reject H₀ if t ≥ 2.012 or t ≤ -2.012.

**3. Decision (We need More Information to Make a Decision)**

To make a decision, we need the following information:

* **Sample mean (x̄):** The average time her class actually took.
* **Sample standard deviation (s):** The standard deviation of the times her class took.

Without this information, we cannot calculate the t-statistic and make a decision.

**Example Scenario (to illustrate the decision-making process):**

Let's assume:

* Sample mean (x̄) = 18 minutes
* Sample standard deviation (s) = 5 minutes

Now, we can calculate the t-statistic:

* t = (x̄ - μ) / (s / √n)
* t = (18 - 20) / (5 / √48)
* t = -2 / (5 / 6.928)
* t = -2 / 0.7216
* t ≈ -2.772

In this example, t = -2.772. Since -2.772 ≤ -2.012, we would reject the null hypothesis.

**4. Interpretation (Based on the Example Scenario)**

If we used the example data, the interpretation would be:

* The results of the t-test indicate that there is a statistically significant difference in the average time taken by the students compared to the teacher's expected 20 minutes.
* Specifically, the students took significantly less time than expected (18 minutes on average in our example).
* This suggests that the teacher's observation of students taking less time is not likely due to chance and that there is a systematic difference.

**Important Note:**

* The actual decision and interpretation depend on the sample mean and standard deviation, which were not provided in the original question.
* If you provide the sample mean and standard deviation, I can give you a more specific answer.