Question

Two 95 percent confidence intervals will be constructed to estimate the difference in means of two populations, R and J. One confidence interval, I400, will be constructed using samples of size 400 from each of R and J, and the other confidence interval, I100, will be constructed using samples of size 100 from each of R and J.When all other things remain the same, which of the following describes the relationship between the two confidence intervals?a)The width of i400 will be 4 times the width of i100.b) The width of i400 will be 2 times the width of i100c)The width of i400 will be equal to the width of i100.d)The width of I400 will be 1/2 times the width of I100e)The width of I400 will be 1/4 times the width of I100.

Answer 1

Answer:

The correct option is (d).

Step-by-step explanation:

The (1 - α)% confidence interval for the difference between two means with same sample size is:

$CI=(\bar x_{1}-\bar x_{2})\pm CV\times SD\times \sqrt{\frac{2}{n}}$

The width of the interval is:

$\text{Width}=2\times CV\times SD\times \sqrt{\frac{2}{n}}$

From the formula of the width of the confidence interval it can be seen that the sample size is inversely related to the width.

That is, if the sample size is increased the width of the interval will be decreased and if the sample size is decreased the width of the interval will be increased.

It is provided that two confidence intervals are constructed for the difference between the means of two populations R and J.

One One confidence interval, will be constructed using samples of size 400 from each of R and J.

And the other confidence interval, will be constructed using samples of size 100 from each of R and J.

Determine the formula of width for both sample sizes as follows:

$\text{Width}_{1}=2\times CV\times SD\times \sqrt{\frac{2}{400}}$

           $=2\times CV\times SD\times \frac{\sqrt{2}}{20}$

$\text{Width}_{2}=2\times CV\times SD\times \sqrt{\frac{2}{100}}$

           $=2\times CV\times SD\times \frac{\sqrt{2}}{10}$

So, the width of I₄₀₀ is half times the width of I₁₀₀.

The correct option is (d).