A random sample of 722 residents in a major town was asked whether they had ever been bitten by a dog. The responses (1=Yes and 2=N0) are recorded. Estimate with 95% confidence the proportion of residents who have been bitten by a dog.
Sample frequencies: n (1) = 304; n (2) = 418
To estimate the proportion of residents who have been bitten by a dog, we can use the formula for the confidence interval of a proportion.
The formula for the confidence interval of a proportion is:
CI = p̂ ± z * √(p̂ * (1 - p̂) / n)
where:
- CI is the confidence interval
- p̂ is the sample proportion (the number of "Yes" responses divided by the total sample size)
- z is the z-score associated with the desired confidence level
- n is the sample size
In this case, we have a sample size of n = 722, and the number of "Yes" responses is 304. The sample proportion is calculated as p̂ = 304 / 722 = 0.421 (rounded to three decimal places).
Since the confidence level is not mentioned in the question, let's assume a 95% confidence level. The corresponding z-score for a 95% confidence level is approximately 1.96.
Using these values, we can now calculate the confidence interval:
CI = 0.421 ± 1.96 * √(0.421 * (1 - 0.421) / 722)
Calculating the values inside the square root:
0.421 * (1 - 0.421) / 722 ≈ 0.000259
CI = 0.421 ± 1.96 * √(0.000259)
Calculating the square root of 0.000259:
√(0.000259) ≈ 0.0161
CI = 0.421 ± 1.96 * 0.0161
Calculating the product of 1.96 and 0.0161:
1.96 * 0.0161 ≈ 0.0316
CI = 0.421 ± 0.0316
Finally, the confidence interval is:
CI = (0.421 - 0.0316, 0.421 + 0.0316)
Simplifying the values:
CI = (0.3894, 0.4526)
Therefore, the 95% confidence interval for the proportion of residents who have been bitten by a dog is approximately 0.3894 to 0.4526.