Final answer:
The 95% confidence interval for the proportion of positive test results is calculated using the sample proportion, standard error, and z-score. This interval allows us to estimate the true population proportion with a stated level of confidence, taking into account the variability between samples but not systematic errors or biases.
Step-by-step explanation:
To calculate the 95% confidence interval for the proportion of positive test results, we'll first need the sample proportion (p) and the sample size (n). From the provided information, we have 500 people surveyed, with 280 owning an automobile. This gives us:
- Sample proportion (p) = 280/500 = 0.56
- Sample size (n) = 500
The standard error (SE) of the sample proportion is calculated using the formula SE = sqrt(p(1-p)/n). Plugging in our values, we get SE = sqrt(0.56 * 0.44 / 500). The z-score corresponding to a 95% confidence interval is 1.96 (since 2.5% is on each side of the normal distribution).
Then, the margin of error (E) is z * SE. Our confidence interval is then:
p ± E = 0.56 ± 1.96 * SE
To interpret this confidence interval in the context of medical condition testing, it indicates that we can be 95% confident that the true proportion of the population with positive test results falls within this range. This does not mean that 95% of the sample data falls within this interval, but rather that if we were to take many samples and calculate intervals in the same way, 95% of them would contain the true population proportion.
If we were discussing test accuracy for a medical condition, the confidence interval would give us an estimate of the true rate at which the test yields a positive result within the population being studied. We would have to consider, however, that the interval does not account for systematic errors or biases in the testing process.