Use the data and confidence level to construct a confidence...

Use the data and confidence level to construct a confidence interval estimate of $p,$ then address the given question. In clinical trials of the drug Lipitor (atorvastatin), 270 subjects were given a placebo and 7 of them had allergic reactions. Among 863 subjects treated with $10 \mathrm{mg}$ of the drug. 8 experienced allergic reactions. Construct the two $95 \%$ confidence interval estimates of the percentages of allergic reactions. Compare the results. What do you conclude?

Use the data and confidence level to construct a confidence interval estimate of $p,$ then address the given question.
In clinical trials of the drug Lipitor (atorvastatin), 270 subjects were given a placebo and 7 of them had allergic reactions. Among 863 subjects treated with $10 \mathrm{mg}$ of the drug. 8 experienced allergic reactions. Construct the two $95 \%$ confidence interval estimates of the percentages of allergic reactions. Compare the results. What do you conclude?

Key Concepts

Comparison of Confidence Intervals

Comparing confidence intervals from different samples can help assess whether there is a significant difference between the estimated proportions. If the intervals overlap considerably, this may indicate that there is no statistically significant difference between the underlying probabilities of the event in the two groups.

Confidence Interval for a Proportion

This involves computing an interval estimate that, with a certain level of confidence (such as 95%), is believed to contain the true population proportion. The interval is typically calculated as the sample proportion plus or minus the product of a critical value (from the normal distribution) and the standard error, which is a function of the sample proportion and size.

Critical Value and Confidence Level

The critical value, such as 1.96 for a 95% confidence level, is used in constructing the confidence interval. It represents the number of standard deviations needed to capture the central percentage (95% in this case) of a normal distribution, and thus it plays a crucial role in determining the width of the confidence interval.

Binomial Proportion

This concept refers to the parameter p of a binomial distribution, which represents the probability of success (or occurrence of an event, such as an allergic reaction) in a single trial. The estimation of a binomial proportion is based on the observed rate (or sample proportion) in a set of binary data, and it is fundamental in settings where outcomes are classified in two categories, like reaction or no reaction.

Normal Approximation to the Binomial

When the sample size is sufficiently large, the distribution of the sample proportion can be approximated by a normal distribution. This approximation, justified by the Central Limit Theorem, allows for the construction of confidence intervals around the estimated proportion using the normal distribution's properties.

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