Question

after a set of surveys was compiled and analyzed, the mean value was reported as \\(5.1 \pm 0.4\\).

select the true statement for this data.

a.) the estimate is equal to \\(\pm 0.4\\) and the confidence interval is \\(4.7\\) to \\(5.5\\).

b.) the margin of error is equal to \\(5.1\\) and the estimate is equal to \\(\pm 0.4\\).

c.) the estimate is equal to \\(5.1\\) and the confidence interval is \\(4.7\\) to \\(5.5\\).

d.) the margin of error is equal to \\(\pm 0.4\\) and the confidence interval is \\(5.1\\).

Explanation:

Identify the components of the reported value

The problem states that the mean value is reported as \(5.1 \pm 0.4\).
In statistical reporting, an interval estimate is typically expressed in the format:

$$\text{Estimate} \pm \text{Margin of Error}$$

Here, the point estimate (the sample mean) is \(5.1\), and the margin of error is \(0.4\) (or represented as \(\pm 0.4\)).

Calculate the confidence interval boundaries

To find the confidence interval, we subtract and add the margin of error to the point estimate:

• Lower limit: \(5.1 - 0.4 = 4.7\)
• Upper limit: \(5.1 + 0.4 = 5.5\)

Thus, the confidence interval ranges from \(4.7\) to \(5.5\).

Evaluate the given options

Let's analyze each statement:

• a.) "The estimate is equal to \(\pm 0.4\) and the confidence interval is \(4.7\) to \(5.5\)." (Incorrect: The estimate is \(5.1\), not \(\pm 0.4\)).
• b.) "The margin of error is equal to \(5.1\) and the estimate is equal to \(\pm 0.4\)." (Incorrect: The margin of error is \(0.4\) and the estimate is \(5.1\)).
• c.) "The estimate is equal to \(5.1\) and the confidence interval is \(4.7\) to \(5.5\)." (Correct: The point estimate is \(5.1\), and the interval is indeed \(4.7\) to \(5.5\)).
• d.) "The margin of error is equal to \(\pm 0.4\) and the confidence interval is \(5.1\)." (Incorrect: The confidence interval is a range, not a single value).

Answer:

• a.) The estimate is equal to \(\pm 0.4\) and the confidence interval is 4.7 to 5.5.
• b.) The margin of error is equal to 5.1 and the estimate is equal to \(\pm 0.4\).
• c.) The estimate is equal to 5.1 and the confidence interval is 4.7 to 5.5. (Correct answer)
• d.) The margin of error is equal to \(\pm 0.4\) and the confidence interval is 5.1.