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 i100
c)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 I100
e)The width of I400 will be 1/4 times the width of I100.

Answer 1

Answer:

e)The width of I400 will be 1/4 times the width of I100.

Step-by-step explanation:

If the confidence interval is 95% = 0.95, therefore α = 1 - 0.95 = 0.05.

$\frac{\alpha }{2} =\frac{0.05}{2}=0.0025$

The width of the confidence (W) is given by the equation:

$W=\frac{z_{\frac{\alpha }{2} }*\sigma}{\sqrt{n} }$,  where n is the sample size, $\sigma$ is the standard deviation and $z_{\frac{\alpha }{2} }$ is the z score of α/2

From the equation, we can see that the width is inversely proportional to the square root of the sample size.

Since nterval, 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. and  all other things remain the same.

Let $W_{I4$ be the width of I400 and $W_{I1$ be the width of I100. Therefore:

$W_{I4}=\frac{z_{\frac{\alpha }{2} }*\sigma}{\sqrt{400} }=\frac{z_{\frac{\alpha }{2} }*\sigma}{4 }\\And, W_{I1}=\frac{z_{\frac{\alpha }{2} }*\sigma}{\sqrt{100} }=\frac{z_{\frac{\alpha }{2} }*\sigma}{1 }\\Therefore,\frac{W_{I4}}{W_{I1}} =\frac{\frac{z_{\frac{\alpha }{2} }*\sigma}{4 }}{\frac{z_{\frac{\alpha }{2} }*\sigma}{1 }}\\\frac{W_{I4}}{W_{I1}}=\frac{1}{4}\\ W_{I4}=\frac{1}{4}W_{I1}$

The width of I400 will be 1/4 times the width of I100.

Answer 2

Answer:

d)The width of I400 will be 1/2 times the width of I100

Step-by-step explanation:

i do believe that the answer to this question is D because of the square root in the denominator of the standard error formula. When the square root of the sample size is taken, 400 becomes 20 and the square root of 100 becomes 10.

this can lead us to believe that I400 will be 1/2 times the width of I100. we are dividing by a larger number when taking the square root of 400, so that width will be smaller than the 100 size sample.