The 97% confidence interval for the population proportion of smokers is approximately 4.8% to 9.2%.
Constructing a confidence interval for the population proportion involves using statistical methods to estimate the range within which the true proportion is likely to fall. In this scenario, we are interested in estimating the proportion of smokers in a population. The confidence interval is constructed based on a random sample of 500 people, where 32 individuals are identified as smokers.
The formula for the confidence interval for a population proportion is derived from the normal approximation to the binomial distribution. The formula includes the sample proportion (p^), the Z-score corresponding to the desired confidence level (in this case, 97%), and the square root of a term involving the sample proportion and the sample size.
The sample proportion (p^) is calculated by dividing the number of individuals with a certain characteristic (smokers) by the total sample size. The Z-score is determined based on the desired confidence level, and it represents the number of standard deviations from the mean in a standard normal distribution.
Substituting the values into the formula yields a confidence interval, which provides a range of values within which we can be reasonably confident that the true population proportion of smokers lies. In this case, the 97% confidence interval for the proportion of smokers is approximately 4.8% to 9.2%. This implies that if we were to obtain multiple samples and construct confidence intervals for each, we would expect about 97% of these intervals to contain the true population proportion of smokers.