Question

Suppose that a 95% confidence interval states that population mean is greater than 100 and less than 300. Then the value of sample mean \((\bar{x})\) and margin of error (E) respectively are :

• A. 150, ±100
• B. 100, ±100
• C. 250, ±50
• D. 200, ±100

Answer 1

The Correct Option is D

To find the sample mean and the margin of error, we start by identifying the boundaries of the 95% confidence interval given as greater than 100 and less than 300. These are the endpoints of the interval which can be represented as [100, 300].

The sample mean \(\bar{x}\) is the midpoint of the confidence interval. It is calculated using the formula:

\[ \bar{x} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2} \]

Substituting the given bounds:

\[ \bar{x} = \frac{100 + 300}{2} = \frac{400}{2} = 200 \]

The margin of error (E) is the difference between the sample mean and either endpoint of the confidence interval. It can be calculated as:

\[ E = \text{Upper Bound} - \bar{x} \]

or equivalently:

\[ E = \bar{x} - \text{Lower Bound} \]

Using the sample mean:

\[ E = 300 - 200 = 100 \]

So, the sample mean \(\bar{x}\) is 200 and the margin of error E is ±100.