Question

6.3 (1 point) Fireworks. Last summer, Survey USA published results of a survey stating that 337 of 571 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 90 % 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 90 % 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 4.6%. 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

1)

point estimate of sample proportion, = 337/571 = 0.5902

2)

sample size, n = 571
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.5902 * (1 - 0.5902)/571) = 0.0206

3)

Given CI level is 90%, hence α = 1 - 0.9 = 0.1
α/2 = 0.1/2 = 0.05, Zc = Z(α/2) = 1.64

Margin of Error, ME = zc * SE
ME = 1.64 * 0.0206
ME = 0.0338

4)

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.5902 - 1.64 * 0.0206 , 0.5902 + 1.64 * 0.0206)
CI = (0.5564 , 0.6240)

5)
The following information is provided,
Significance Level, α = 0.01, Margin of Error, E = 0.046

The provided estimate of proportion p is, p = 0.5902
The critical value for significance level, α = 0.01 is 2.576.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.5902*(1 - 0.5902)*(2.576/0.046)^2
n = 758.49

Therefore, the sample size needed to satisfy the condition n >= 758.49 and it must be an integer number, we conclude that the minimum required sample size is n = 759
Ans : Sample size, n = 759 or 758