In a survey of 1037 adults from the United States age 65 and over, 643 were concerned about getting the flu. Construct a 90% confidence interval for the population proportion.
To construct a confidence interval for a population proportion, we can use the formula:
CI = p̂ ± Z * √((p̂(1-p̂)) / n)
Where:
- CI represents the confidence interval
- p̂ is the sample proportion (in this case, it's the proportion of adults concerned about getting the flu, which is 643/1037)
- Z is the z-score corresponding to the desired confidence level (90% confidence level corresponds to a z-score of 1.645)
- n is the sample size (in this case, it is 1037)
Now, let's compute the confidence interval step by step:
Step 1: Calculate the sample proportion (p̂):
p̂ = 643/1037 ≈ 0.6199 (rounded to four decimal places)
Step 2: Determine the z-score corresponding to the desired confidence level (90%):
The z-score for a 90% confidence level is 1.645.
Step 3: Compute the standard error:
Standard error = √((p̂(1-p̂)) / n)
Step 4: Calculate the margin of error (ME):
ME = Z * standard error
Step 5: Find the lower and upper bounds of the confidence interval:
Lower bound = p̂ - ME
Upper bound = p̂ + ME
Let's plug in the values into the formula to construct the confidence interval:
Standard error = √((0.6199(1-0.6199)) / 1037) ≈ 0.0148 (rounded to four decimal places)
ME = 1.645 * 0.0148 ≈ 0.0243 (rounded to four decimal places)
Lower bound = 0.6199 - 0.0243 ≈ 0.5956 (rounded to four decimal places)
Upper bound = 0.6199 + 0.0243 ≈ 0.6442 (rounded to four decimal places)
Therefore, the 90% confidence interval for the population proportion of adults concerned about getting the flu is approximately 0.5956 to 0.6442.