Find a 95% confidence interval for the true population proportion.
In a survey of 415 adults,59% said that they favored the proposed environmental laws
To find a 95% confidence interval for the true population proportion, we can use the formula:
CI = p̂ ± Z * √(p̂(1 - p̂) / n)
where:
CI = confidence interval
p̂ = sample proportion
Z = z-score corresponding to the desired confidence level
n = sample size
In this case, the sample proportion (p̂) is 59% or 0.59, and the sample size (n) is 415.
First, we need to find the z-score for a 95% confidence level. The z-score for a 95% confidence level is approximately 1.96.
CI = 0.59 ± 1.96 * √(0.59(1 - 0.59) / 415)
Calculating the expression inside the square root:
p̂(1 - p̂) = 0.59(1 - 0.59) = 0.59(0.41) = 0.2419
Now, substitute the values into the formula:
CI = 0.59 ± 1.96 * √(0.2419 / 415)
Calculating the square root:
√(0.2419 / 415) ≈√ 0.000583855322 ≈ 0.0241499123
Now substitute back into the formula:
CI = 0.59 ± 1.96 * 0.0241499123
Calculating the multiplication:
1.96 * 0.0241499123 ≈ 0.047354638
Finally, substitute into the formula:
CI = 0.59 ± 0.047354638
Calculating the upper and lower limits separately:
Upper limit = 0.59 + 0.047354638 ≈ 0.63735
Lower limit = 0.59 - 0.047354638 ≈ 0.54265
Therefore, the 95% confidence interval for the true population proportion is approximately 0.54265 to 0.63735 or (0.54265, 0.63735).