Question

According to a survey of 737 small business participants in the Constant Contact Small Biz Council in May of 2013, 402 of the respondents say it is harder to run a small business now than it was 5 years ago. When estimating the population proportion, what is the 99% confidence interval estimating the proportion of businesses who believe it is harder to run a business now than 5 years ago?

1) ( -0.49821 , 0.5927 )

2) ( 0.49821 , 0.5927 )

3) ( 0.4073 , 0.50179 )

4) ( 0.50279 , 0.58812 )

5) ( 0.52711 , 0.5638 )

Answer 1

Solution :

Given that,

n = 737

x = 402

= x / n = 402 / 737 = 0.545

1 - = 1 - 0.545 = 0.455

At 99% confidence level the z is ,

= 1 - 99% = 1 - 0.99 = 0.01

/ 2 = 0.01 / 2 = 0.005

Z_/2 = Z_0.005 = 2.576

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 2.576 * (((0.545 * 0.455) / 737)

= 0.04679

A 99 % confidence interval for population proportion p is ,

- E < P < + E

0.545 - 0.0477 < p < 0.545 + 0.04679

0.49821 < p < 0.5927

( 0.4921, 0.5927)

Option 2 ) is correct.