Question

A market survey with a sample size of 1000 was conducted for a parameter that follows normal distribution. The confidence interval was estimated as [500, 700] with a mean of 600. It is now desired to reduce the confidence interval to [550, 650]. The sample size for achieving the desired interval at the same confidence level is

• A. 1000
• B. 4000
• C. 9000
• D. 16000

Hint:

For the same confidence level, halving the confidence interval width requires four times the sample size.

Answer 1

The Correct Option is B

The confidence interval width depends on sample size according to the relation: \[ \text{CI width} \propto \frac{1}{\sqrt{n}} \]

Step 1: Determine the original and desired half-widths.
Original interval: [500, 700] Half-width: \[ E_1 = 700 - 600 = 100 \] Desired interval: [550, 650] Half-width: \[ E_2 = 650 - 600 = 50 \]

Step 2: Use the proportionality relation.
\[ \frac{E_2}{E_1} = \sqrt{\frac{n_1}{n_2}} \] Given: \(n_1 = 1000\), \(E_1 = 100\), \(E_2 = 50\). \[ \frac{50}{100} = \sqrt{\frac{1000}{n_2}} \] \[ \frac{1}{2} = \sqrt{\frac{1000}{n_2}} \] Square both sides: \[ \frac{1}{4} = \frac{1000}{n_2} \] \[ n_2 = 4000 \]

Step 3: Conclusion.
To cut the confidence interval width in half, the sample size must be quadrupled: \[ n_2 = 4 \times 1000 = 4000 \]

Final Answer: (B) 4000