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
[tex]\alpha = (100 - 95 ) \%[/tex]
=> [tex]\alpha = 0.05[/tex]
Generally from the normal distribution table the critical value of [tex]\frac{\alpha }{2}[/tex] is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
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
[tex]E = Z_{\frac{\alpha }{2} } * \frac{\sigma}{\sqrt{n} }[/tex]
Here assume that [tex]Z_{\frac{\alpha }{2} } \ and \ \sigma \[/tex] is constant so
[tex]E = \frac{k}{\sqrt{n} }[/tex]
=> [tex]E \sqrt{n} = K[/tex]
=> [tex]E_1 \sqrt{n}_1 = E_2 \sqrt{n}_2[/tex]
So let [tex]n_1 = 400[/tex] and [tex]n_2 = 100[/tex]
=> [tex]E_1 \sqrt{400} = E_2 \sqrt{100}[/tex]
=> [tex]E_1 = \frac{\sqrt{100} }{\sqrt{400} } E_2[/tex]
=> [tex]E_1 = \frac{1}{2 } E_2[/tex]
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
