In a poll of 1,004 adults, 93% indicated that restaurants and bars should refuse service to patrons who have had too much to drink. Construct the 90% confidence interval for the proportion of all adults who feel the same way.
Formula:
CI90 = p + or - 1.645(√pq/n)
Note: q = 1 - p
With your data:
CI90 = .93 + or - 1.645(√(.93)(.07)/1004)
I'll let you take it from here to finish the calculations.
To construct the confidence interval for the proportion of all adults who feel the same way, we will use the formula for estimating a confidence interval for a proportion:
Confidence Interval = Sample Proportion ± Margin of Error
Where:
- Sample Proportion is the proportion of the sample that indicated a certain response (in this case, 93% or 0.93).
- Margin of Error represents the range within which the true population proportion is likely to fall.
To calculate the Margin of Error, we need to determine the standard error of the proportion. The formula for the standard error of the proportion is:
Standard Error = √((p * (1-p)) / n)
Where:
- p is the sample proportion (0.93)
- n is the sample size (1,004)
Let's calculate it step by step:
Step 1: Calculate the Standard Error
Standard Error = √((0.93 * (1-0.93)) / 1004)
= √((0.93 * 0.07) / 1004)
= √(0.0651 / 1004)
= 0.00805 (rounded to five decimal places)
Step 2: Determine the Margin of Error
The Margin of Error is obtained by multiplying the Standard Error by the appropriate critical value. In this case, since we want a 90% confidence interval, we need to determine the critical value that corresponds to a 5% significance level.
The critical value can be found using a Z-score table or calculated using a statistical software. For a 90% confidence level, the critical value is approximately 1.645.
Margin of Error = 1.645 * 0.00805
= 0.013241 (rounded to six decimal places)
Step 3: Calculate the Confidence Interval
To calculate the lower and upper bounds of the confidence interval, we subtract and add the Margin of Error to the Sample Proportion.
Lower Bound = 0.93 - 0.013241
= 0.916759 (rounded to six decimal places)
Upper Bound = 0.93 + 0.013241
= 0.943241 (rounded to six decimal places)
Therefore, the 90% confidence interval for the proportion of all adults who feel the same way is approximately 0.9168 to 0.9432.