To determine the range within which Ardem's actual population mean will lie, let's go through the steps step-by-step:
1. Calculate the Mean of Each Sample:
- For Sample 1: (4 + 5 + 2 + 4 + 3) / 5 = 18 / 5 = 3.6
- For Sample 2: (2 + 2 + 6 + 5 + 7) / 5 = 22 / 5 = 4.4
- For Sample 3: (4 + 6 + 3 + 4 + 1) / 5 = 18 / 5 = 3.6
- For Sample 4: (5 + 2 + 4 + 3 + 6) / 5 = 20 / 5 = 4.0
2. Calculate the Overall Mean of These Sample Means:
- Mean values of the samples are 3.6, 4.4, 3.6, and 4.0.
- Overall mean = (3.6 + 4.4 + 3.6 + 4.0) / 4 = 15.6 / 4 = 3.9
3. Calculate the Standard Error of the Mean:
- Find the standard deviation (SD) of the sample means:
- Mean of sample means = 3.9
- Differences from mean: (3.6 - 3.9), (4.4 - 3.9), (3.6 - 3.9), (4.0 - 3.9)
- Squared differences: (-0.3)^2, (0.5)^2, (-0.3)^2, (0.1)^2 = 0.09, 0.25, 0.09, 0.01
- Variance = (0.09 + 0.25 + 0.09 + 0.01) / (4 - 1) = 0.44 / 3 ≈ 0.147
- SD = sqrt(0.147) ≈ 0.383
- Standard error (SE) = SD / sqrt(number of samples) = 0.383 / sqrt(4) = 0.383 / 2 ≈ 0.191
4. Determine the Confidence Interval:
- Using a 95% confidence level, the critical value (z-score) is approximately 1.96.
- Margin of error = z-score Standard error = 1.96 0.191 ≈ 0.374
- Lower bound of the confidence interval = Overall mean - Margin of error = 3.9 - 0.374 ≈ 3.525
- Upper bound of the confidence interval = Overall mean + Margin of error = 3.9 + 0.374 ≈ 4.275
Therefore, the range within which Ardem's actual population mean will lie is between approximately 3.525 and 4.275. From the given options, the interval that includes this range is:
[tex]\( \boxed{(3.6 \text{ and } 4.4)} \)[/tex]
