In Problems 1-8, construct the appropriate confidence...

In Problems $1-8,$ construct the appropriate confidence interval. A simple random sample of size $n=785$ adults was asked if they follow college football. Of the 785 surveyed, 275 responded that they did follow college football. Construct a $95 \%$ confidence interval for the population proportion of adults who follow college football.

In Problems $1-8,$ construct the appropriate confidence interval.
A simple random sample of size $n=785$ adults was asked if they follow college football. Of the 785 surveyed, 275 responded that they did follow college football. Construct a $95 \%$ confidence interval for the population proportion of adults who follow college football.

Key Concepts

Margin of Error

The margin of error quantifies the uncertainty in the estimate of a population parameter. It is computed by multiplying a critical value (from the z-distribution for a specified confidence level) with the standard error of the estimate. A smaller margin of error indicates a more precise estimate of the parameter.

Confidence Interval

A confidence interval provides a range of plausible values for a population parameter based on sample data. It is constructed using the point estimate from the sample, a critical value from a probability distribution (often the z-distribution for large samples), and the standard error. The interval reflects the level of confidence, such as 95%, that it contains the true parameter.

Population Proportion

The population proportion is the fraction of the population that has a particular attribute or characteristic. In contexts such as survey results, it represents the true prevalence of a condition in the overall population, and the sample proportion is used as an estimate for this value.

Sampling Distribution

The sampling distribution of the sample proportion describes the probability distribution of the sample proportion across many possible samples from the population. For large sample sizes, the Central Limit Theorem ensures that this distribution is approximately normal, which is key to applying normal-based inference methods like constructing confidence intervals.

Standard Error for Proportion

The standard error measures the variability of the sample proportion estimate due to random sampling. It is calculated using the formula sqrt[(p?(1 - p?))/n], where p? is the sample proportion and n is the sample size. This standard error is essential for determining the margin of error in a confidence interval.

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