Construct the appropriate confidence interval. A simple...

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 about the population proportion of adults who follow college football.

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 about the population proportion of adults who follow college football.

Key Concepts

Margin of Error

The margin of error represents the range of uncertainty around the sample estimate. It is calculated as the product of the critical value and the standard error, and it is added to and subtracted from the sample statistic to obtain the confidence interval.

Confidence Interval

A confidence interval is a range derived from sample data that is believed to cover the true population parameter with a specified level of confidence. It quantifies the uncertainty associated with the sample statistic and gives a range within which the actual population value is expected to fall.

Sample Proportion

The sample proportion is an estimate of the population proportion, calculated as the number of successes divided by the total number of observations in a sample. It serves as a basis for making inferences about the corresponding population parameter.

Standard Error

The standard error for a proportion measures the dispersion of the sample proportion around the true population proportion. It is calculated using the formula sqrt[p(1-p)/n], where p is the sample proportion and n is the sample size, and it reflects the variability due to random sampling.

Critical Value

The critical value is a threshold from the standard normal or t-distribution corresponding to the desired confidence level (such as 95%). This value is used to determine the margin of error by scaling the standard error.

Simple Random Sampling

Simple random sampling is a method of selecting a sample in which every member of the population has an equal chance of being chosen. This technique helps ensure that the sample is representative of the population, thereby supporting the validity of statistical inferences made using the sample data.

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