Find the indicated confidence interval. Assume the standard...

Find the indicated confidence interval. Assume the standard error comes from a bootstrap distribution that is approximately normally distributed.
A $99 \%$ confidence interval for a proportion $p$ if the sample has $n=200$ with $\hat{p}=0.78,$ and the standard error is $S E=0.03$.

Key Concepts

Confidence Interval

A confidence interval is a range of values derived from sample data that is likely to contain the true value of an unknown population parameter. It is calculated by taking the point estimate and adding and subtracting a margin of error, which is determined by the standard error and a critical value from a probability distribution corresponding to the desired confidence level.

Standard Error

The standard error quantifies the amount of variability or dispersion in the sampling distribution of a statistic, such as a sample proportion or sample mean. It is used to measure how much the estimate of the parameter would fluctuate from sample to sample, making it crucial for constructing confidence intervals.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, indicating that data near the mean are more frequent in occurrence. In the context of confidence intervals, when the sampling distribution of an estimator is approximately normal, even for non-normally distributed raw data, it allows for the use of critical values from the normal distribution to compute the interval.

Bootstrap Distribution

A bootstrap distribution is an empirical distribution of a statistic obtained by resampling with replacement from the observed data. This non-parametric method allows for estimation of the standard error and other properties of the estimator without relying on strict parametric assumptions about the underlying population distribution.

Proportion

In statistics, the proportion is a measure that expresses the fraction or percentage of the total that exhibits a particular attribute. In many applications such as surveys or experiments, the sample proportion is used as an estimate of the true population proportion, and its variability is measured using standard error to construct confidence intervals.

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