SOLUTION: a government sponsored telephone counseling service for adolescents tested whether the length of calls would be affected by a special telephone system that had a better sound quali

Question 1185740: a government sponsored telephone counseling service for adolescents tested whether the length of calls would be affected by a special telephone system that had a better sound quality.Over the past several years, the legths of telephone calls (in minutes) were normally distributed with Population M = 18, and Population SD = 8. They arranged to have the special phone system loaned to them for one day. On that day, the mean length of the 46 calls they received was 21 minutes. Test whether the length of calls has changed using the .05 signifance level (A) Carry out the Z-test using the 5 steps of hypothesis testing. Please include the Pop M, Pop SD, N, M, one tail or two tail, Mean Population Mm, Population SD2m, Population SDm
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
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

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