Question 1185740
Here's how to conduct a z-test for this hypothesis, following the five steps:

**(A) Z-test for Call Lengths with Improved Sound Quality**

**1. State the Hypotheses:**

*   **Null Hypothesis (H0):** The improved sound quality has no effect on call length.  μ = 18
*   **Alternative Hypothesis (H1):** The improved sound quality *does* affect call length. μ ≠ 18 (This is a two-tailed test because we're testing for a *change*, not specifically an increase or decrease).

**2. Set the Criteria for a Decision:**

*   **Significance Level (alpha):** α = 0.05
*   **Critical Values:** Since it's a two-tailed test, we divide alpha by 2 (0.05 / 2 = 0.025) and look up the corresponding z-scores in both tails of the standard normal distribution. The critical values are approximately ±1.96.
*   **Decision Rule:** Reject H0 if the calculated z-score is greater than +1.96 *or* less than -1.96.

**3. Compute the Test Statistic:**

```
z = (M - μ) / σM
```

Where:

*   M = Sample mean = 21
*   μ = Population mean = 18
*   σM = Standard error of the mean = σ / √N
*   σ = Population standard deviation = 8
*   N = Sample size = 46

First, calculate the standard error of the mean (σM):

```
σM = 8 / √46 ≈ 8 / 6.78 ≈ 1.18
```

Now, calculate the z-score:

```
z = (21 - 18) / 1.18
z ≈ 2.54
```

**4. Make a Decision:**

*   **Comparison:** The calculated z-score (2.54) is greater than the positive critical value (1.96).
*   **Decision:** We *reject* the null hypothesis.

**5. State the Conclusion:**

There is sufficient evidence at the 0.05 significance level to conclude that the improved sound quality *does* affect the length of telephone calls.

**Summary of Values:**

*   Population M (μ): 18
*   Population SD (σ): 8
*   N: 46
*   M (Sample Mean): 21
*   One-tailed or Two-tailed: Two-tailed
*   Population Mean of the Sampling Distribution (μM): 18 (same as population mean)
*   Population Variance of the Sampling Distribution (σ²M): σ²/N = 8²/46 = 64/46 ≈ 1.39
*   Standard Deviation of the Sampling Distribution (σM): σ/√N = 8/√46 ≈ 1.18