Question

Use the applet "Sample Size and Interval Width when Estimating Proportions" to answer the following questions.

This applet illustrates how sample size is related to the width of a 95% confidence interval estimate for a population proportion.

(a)

At 95% confidence, how large a sample should be taken to obtain a margin of error of 0.023?

(b)

As the sample size decreases for any given confidence level, what happens to the confidence interval?

The width of the confidence interval becomes the same as the standard error.The confidence interval becomes more narrow because the sampling distribution becomes larger.    The confidence interval becomes wider because the standard error becomes larger.The confidence interval becomes wider than the population proportion.The confidence interval becomes more narrow than the population proportion.

Answer 1

a )

We have to find Sample size ( n )

Margin of Error =E = 0.023

Formula :

Where ,

p is previous estimate of population proportion.

{ If p is not given take it as 0.5 }

is critical value for given confidence level.

Confidence level = 95%

Significance level = = 1 - 0.95 = 0.05   , /2 = 0.025

So critical value is ,  

= = 1.96         { using Excel , =NORMSINV(1-0.025 ) = 1.96 }

So, sample size n is ,

Sample of size n = 1816 should be taken to obtain a margin of error of 0.023

b)

Formula of standard Error of proportion is ,

n is in denominator. So if n increases then SE decrease and if n decreases then SE increases.

Therefore, If sample size decreases then confidence interval becomes wider because the standard error becomes larger .