Question

A 95% confidence interval for a population mean is computed from a sample of size 400. Another 95% confidence interval will be computed from a sample of size 100 drawn from the same population. Choose the best answer to fill in the blank: The interval from the sample of size 400 will be approximately _____ as the interval from the sample of size 100. Multiple Choice

Answer 1

Answer:

The interval from the sample of size 400 will be approximately One -half as wide as the interval from the sample of size 100

Step-by-step explanation:

From the question we are told the confidence level is  95% , hence the level of significance is    

      $\alpha = (100 - 95 ) \%$

=>   $\alpha = 0.05$

Generally from the normal distribution table the critical value  of  $\frac{\alpha }{2}$ is  

   $Z_{\frac{\alpha }{2} } = 1.96$

Generally the 95% confidence interval is dependent on the value of the margin of error at a constant sample mean or sample proportion

Generally the margin of error is mathematically represented as

      $E = Z_{\frac{\alpha }{2} } * \frac{\sigma}{\sqrt{n} }$    

Here assume that $Z_{\frac{\alpha }{2} } \ and \ \sigma$ is constant so

     $E = \frac{k}{\sqrt{n} }$

=>  $E \sqrt{n} = K$

=>   $E_1 \sqrt{n}_1 = E_2 \sqrt{n}_2$

So  let  $n_1 = 400$ and $n_2 = 100$

=>   $E_1 \sqrt{400} = E_2 \sqrt{100}$

=>  $E_1 = \frac{\sqrt{100} }{\sqrt{400} } E_2$

=>  $E_1 = \frac{1}{2 } E_2$

So From this we see that  the confidence interval for a sample size of 400 will be half that with a sample size of 100