Final answer:
The 95% confidence interval for the proportion of adults aged 35 to 45 considering participation in extreme sports is [19.3%, 24.7%], meaning there is a 95% certainty that the true proportion lies within this range.
Step-by-step explanation:
To find the 95% confidence interval for the proportion of adults ages 35 to 45 who have thought about participating in an extreme sport such as bungee jumping, we first use the point estimate provided by the sample. In this case, 22% of the surveyed 900 adults have thought about this, which gives us a point estimate (p) of 0.22. We also need the standard error (SE) for the proportion, which can be calculated using the formula SE = √[p(1-p)/n], where n is the sample size.
Calculating SE gives us:
SE = √[0.22(1-0.22)/900] ≈ √[0.1716/900] ≈ √[0.0001906667] ≈ 0.013806.
To construct the 95% confidence interval, we use the Z-score for a 95% confidence level, which is commonly 1.96 for a normal distribution. The margin of error (E) is calculated as:
E = Z⋅SE = 1.96⋅0.013806 ≈ 0.02705976.
The 95% confidence interval is then:
p ± E = 0.22 ± 0.02705976,
which gives us the interval [0.19294024, 0.24705976], after rounding to the nearest tenth of a percent, we have [19.3%, 24.7%].
The correct interpretation of this confidence interval is that we are 95% confident that the true proportion of all adults ages 35 to 45 who have thought about participating in an extreme sport lies between 19.3% and 24.7%.