A simple random sample of 300 branches out of a total of...

A simple random sample of 300 branches out of a total of 1200 branches of a UK travel agency found that 75 had at least one staff member over the age of 55 . Find a $95 \%$ confidence interval for the proportion of all the branches having a staff member over 55 .

Key Concepts

Simple Random Sampling

Simple random sampling is a sampling method where each member of the population has an equal chance of being selected. This method helps to ensure that the sample is representative of the population, which is crucial for generalizing the findings, such as estimating a population proportion with confidence intervals.

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. In the context of proportions, the confidence interval provides an estimated range for the true proportion, taking into account the variability inherent in the sampling process and a chosen level of confidence.

Proportion

Proportion refers to the fraction of the total that exhibits a particular characteristic, often represented as a percentage or a value between 0 and 1. When working with sample data, the sample proportion serves as a point estimate of the true proportion in the population.

Standard Error

The standard error for a proportion quantifies the variability of the sample proportion from sample to sample. It is calculated using the formula SE = sqrt[(p*(1 - p)) / n], where p is the sample proportion and n is the sample size, and it is essential for determining the width of the confidence interval.

Z-Critical Value

The z-critical value corresponds to the desired level of confidence and is drawn from the standard normal distribution. For example, a 95% confidence level typically uses a z-critical value of approximately 1.96. This value is used to scale the standard error in the construction of the confidence interval.

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