Question

Construct the confidence interval specified to estimate \(\mu\). Arrange the intervals from lowest to highest based on length of the interval:
A. \(95\%\) confidence for \(\bar{x}=20,\ \sigma=3,\ n=49\);
B. \(95\%\) confidence for \(\bar{x}=20,\ \sigma=3,\ n=81\);
C. \(95\%\) confidence for \(\bar{x}=115,\ \sigma=25,\ n=49\);
D. \(95\%\) confidence for \(\bar{x}=115,\ \sigma=25,\ n=81\).

• A. \(A,C,B,D\)
• B. \(D,B,C,A\)
• C. \(B,A,D,C\)
• D. \(A,B,C,D\)

Hint:

For same confidence level, interval length increases with \(\sigma\) and decreases with \(\sqrt{n}\).

Answer 1

The Correct Option is C

Concept:
Length of confidence interval depends on: \[ \frac{\sigma}{\sqrt{n}} \] For same confidence level, compare: \[ \frac{\sigma}{\sqrt{n}} \]

Step 1: Calculate proportional length for A. \[ \frac{3}{\sqrt{49}}=\frac{3}{7} \]

Step 2: For B. \[ \frac{3}{\sqrt{81}}=\frac{3}{9} \]

Step 3: For C. \[ \frac{25}{\sqrt{49}}=\frac{25}{7} \]

Step 4: For D. \[ \frac{25}{\sqrt{81}}=\frac{25}{9} \]

Step 5: Arrange from shortest to longest. \[ \frac{3}{9}<\frac{3}{7}<\frac{25}{9}<\frac{25}{7} \] So: \[ B,A,D,C \] \[ \therefore \text{Correct Answer is (C)} \]