Use the confidence interval to find the margin of error and the sample mean. left parenthesis 0.178 comma 0.360 right parenthesis
The correct answer and explanation is:
To find the margin of error and the sample mean from the confidence interval (0.178,0.360)(0.178, 0.360), follow these steps:
1. Sample Mean:
The sample mean is the midpoint of the lower and upper bounds of the confidence interval. To find the sample mean, you average the two values: Sample mean=Lower bound+Upper bound2\text{Sample mean} = \frac{\text{Lower bound} + \text{Upper bound}}{2}
Substitute the values from the interval (0.178,0.360)(0.178, 0.360): Sample mean=0.178+0.3602=0.5382=0.269\text{Sample mean} = \frac{0.178 + 0.360}{2} = \frac{0.538}{2} = 0.269
So, the sample mean is 0.2690.269.
2. Margin of Error:
The margin of error is the distance from the sample mean to either the lower or upper bound of the confidence interval. To find the margin of error, subtract the sample mean from either the upper or lower bound. Both will give the same result: Margin of error=Upper bound−Sample mean=0.360−0.269=0.091\text{Margin of error} = \text{Upper bound} – \text{Sample mean} = 0.360 – 0.269 = 0.091
Alternatively, you could subtract the lower bound from the sample mean: Margin of error=Sample mean−Lower bound=0.269−0.178=0.091\text{Margin of error} = \text{Sample mean} – \text{Lower bound} = 0.269 – 0.178 = 0.091
Thus, the margin of error is 0.0910.091.
Summary:
- Sample mean: 0.269
- Margin of error: 0.091
The margin of error is a measure of how much the sample estimate is expected to vary from the true population value, and the sample mean is the central value that represents the best estimate of the population mean. In this case, the confidence interval suggests that the true population mean is likely to fall within 0.269 ± 0.091.