Question

(1 point) Fireworks. Last summer, Survey USA published results of a survey stating that 301 of 538 randomly sampled Kansas residents planned to set off fireworks on July 4th. Round all results to 4 decimal places.

1. Calculate the point estimate for the proportion of Kansas residents that planned to set off fireworks on July 4th

2. Calculate the standard error for the point estimate you calculated in part 1.

3. Calculate the margin of error for a 99 % confidence interval for the proportion of Kansas residents that planned to set off fireworks on July 4th.

4. What are the lower and upper limits for the 99 % confidence interval. ( , )

5. Use the information from Survey USA poll to determine the sample size needed to construct a 99% confidence interval with a margin of error of no more than 2.7%. For consistency, use the reported sample proportion for the planning value of p* (rounded to 4 decimal places) and round your Z-value to 3 decimal places. Your answer should be an integer.

Answer 1

Solution :

Given that,

n = 538

x = 301

1) Point estimate = sample proportion = = x / n = 301 / 538 = 0.5595

1 - = 1 - 0.5595 = 0.4405

2) =  [p ( 1 - p ) / n] =   [(0.5595 * 0.4405) / 538 ] = 0.0214

At 99% confidence level

= 1 - 99%

=1 - 0.99 =0.01

/2 = 0.005

Z/2 = Z0.005 = 2.576

3) Margin of error = E = Z / 2 *

= 2.576 * 0.0214

= 0.0551

4) A 99% confidence interval for population proportion p is ,

± E  

= 05595  ± 0.0551

= ( 0.5044, 0.6146 )

lower limit = 0.5044

upper limit = 0.6146

5) Margin of error = E = 0.027

sample size = n = (Z / 2 / E )2 * * (1 - )

= (2.576 / 0.027)2 * 0.5595 * 0.4405

= 2243.41

sample size = n = 2244