Question

The Gallup Organization conducted a telephone survey on attitudes toward AIDS (Gallup Monthly, 1991). A total of 1014 individuals were contacted. Each individual was asked whether they agreed with the following statement: "Landlords should have the right to evict a tenant from an apartment because that person has AIDS." One hundred one individuals in the sample agreed with this statement. Use these data to construct a \(90 \%\) confidence interval for the proportion who are in agreement with this statement. Give an interpretation of your interval.

Short answer

The 90% confidence interval for the population proportion who agree with the statement is approximately (0.083, 0.116). This means that with 90% certainty, the proportion of individuals who agree with the statement in the population lies between 8.3% and 11.6%.

Step-by-step solution

Calculation of sample proportion

The goal in the first step is to calculate the sample proportion (\(p\)). The formula to calculate the sample proportion \(p\) is given as \(p = \frac{x}{n}\), where \(x\) is the number of successes and \(n\) is the total number of trials. In this scenario, the number agreeing with the statement (successes) is 101 and the total number of individuals surveyed (trials) is 1014. Substituting these values into the formula yields the approximate proportion of 0.0996.

Calculation of standard error

The standard error (\(SE\)) formula for the proportions-based case is \(SE = \sqrt{\frac{p(1 - p)}{n}}\). Substituting the previously calculated value for \(p\) and the known value for \(n\), the standard error for this sample is approximately 0.0097.

Calculation of z-score

For a 90% confidence interval, the z-score (which represents how many standard deviations away from the mean the value is) is found in a standard normal distribution table. The z-value is 1.645.

Calculation of margin of error

After having the z-score and standard error, the margin of error can be calculated using formula \(ME = z \times SE\). Substituting known values, the calculated margin of error is approximately 0.016.

Calculation of confidence interval

Now to form the 90% confidence interval, subtract the margin of error from the sample proportion for the lower limit and add the margin of error to the sample proportion for the upper limit. This interval ranges approximately from 0.083 to 0.116.

Interpretation of interval

The final step is to interpret the interval. It can be concluded that there's a 90% confidence that the true population proportion of individuals who agree with the statement lies between 8.3% and 11.6%.